Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 637375 20 pageshttpdxdoiorg1011552013637375
Research ArticleA Quantum Mermin-Wagner Theorem fora Generalized Hubbard Model
Mark Kelbert12 and Yurii Suhov234
1 Swansea University Singleton Park Swansea SA2 8PP UK2 Instituto de Mathematica e Estatistica USP Rua de Matao 1010 Cidada Universitaria 05508-090 Sao Paulo SP Brazil3 Statistical Laboratory University of Cambridge Wilberforce Road Cambridge CB3 0WB UK4 IITP RAS Bolshoy Karetny per 18 Moscow 127994 Russia
Correspondence should be addressed to Mark Kelbert markkelbertgmailcom
Received 20 March 2013 Accepted 14 May 2013
Academic Editor Christian Maes
Copyright copy 2013 M Kelbert and Y Suhov This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs withcontinuous spins in the spirit of the Mermin-Wagner theorem In the model considered here the phase space of a single spinis H
1= L
2(119872) where 119872 is a 119889-dimensional unit torus 119872 = R119889
Z119889 with a flat metric The phase space of 119896 spins is H119896=
Lsym2
(119872
119896) the subspace of L
2(119872
119896) formed by functions symmetric under the permutations of the arguments The Fock space H
= oplus119896=01
H119896yields the phase space of a system of a varying (but finite) number of particles We associate a spaceH ≃ H(119894) with
each vertex 119894 isin Γ of a graph (ΓE) satisfying a special bidimensionality property (Physically vertex 119894 represents a heavy ldquoatomrdquoor ldquoionrdquo that does not move but attracts a number of ldquolightrdquo particles) The kinetic energy part of the Hamiltonian includes (i)minusΔ2 the minus a half of the Laplace operator on 119872 responsible for the motion of a particle while ldquotrappedrdquo by a given atomand (ii) an integral term describing possible ldquojumpsrdquo where a particle may join another atom The potential part is an operator ofmultiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials 119880(1)
(119909)119909 isin 119872 describing a field generated by a heavy atom (b) two-body potentials119880(2)
(119909 119910) 119909 119910 isin 119872 showing the interaction betweenpairs of particles belonging to the same atom and (c) two-body potentials119881(119909 119910) 119909 119910 isin 119872 scaled along the graph distance d(119894 119895)between vertices 119894 119895 isin Γ which gives the interaction between particles belonging to different atomsThe systemunder considerationcan be considered as a generalized (bosonic) Hubbard model We assume that a connected Lie group G acts on119872 represented bya Euclidean space or torus of dimension 119889
1015840le 119889 preserving the metric and the volume in 119872 Furthermore we suppose that the
potentials119880(1)119880(2) and 119881 are G-invariant The result of the paper is that any (appropriately defined) Gibbs states generated by theabove Hamiltonian is G-invariant provided that the thermodynamic variables (the fugacity 119911 and the inverse temperature 120573) satisfya certain restriction The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the densitymatrices
1 Introduction
11 Basic Facts on Bi-Dimensional Graphs As in [1] wesuppose that the graph (ΓE) has been given with the set ofvertices Γ and the set of edges E The graph has the propertythat whenever edge (119895
1015840 119895
10158401015840) isin E the reversed edge (119895
10158401015840 119895
1015840)
belongs to E as well Furthermore graph (ΓE) is without
multiple edges and has a bounded degree that is the numberof edges (119895 119895
1015840) with a fixed initial or terminal vertex is
uniformly bounded
sup [max (♯ 1198951015840 isin Γ (119895 119895
1015840) isin E
♯ 119895
1015840isin Γ (119895
1015840 119895) isin E) 119895 isin Γ] lt infin
(1)
2 Advances in Mathematical Physics
The bi-dimensionality property is expressed in the bound
0 lt sup [1119899
♯Σ (119895 119899) 119895 isin Γ 119899 = 1 2 ] lt infin (2)
where Σ(119895 119899) stands for the set of vertices in Γ at the graphdistance 119899 from 119895 isin Γ (a sphere of radius 119899 around 119895)
Σ (119895 119899) = 119895
1015840isin Γ d (119895 119895
1015840) = 119899 (3)
(The graph distance d(119895 1198951015840) = dΓE(119895 119895
1015840) between 119895 119895
1015840isin Γ is
determined as theminimal length of a path on (ΓE) joining 119895and 1198951015840)This implies that for any 119900 isin Γ the cardinality ♯Λ(119900 119899)of the ball
Λ (119900 119899) = 119895
1015840isin Γ d (119900 119895
1015840) le 119899 (4)
grows at most quadratically with 119899A justification for putting a quantum system on a graph
can be that graph-like structures become increasingly popu-lar in rigorous StatisticalMechanics for example in quantumgravity Namely see [2ndash4] On the other hand a number ofproperties of Gibbs ensembles do not depend upon ldquoregular-ityrdquo of an underlying spatial geometry
12 A BosonicModel in the Fock Space With each vertex 119894 isin Γ
we associate a copy of a compact manifold119872 which we takein this paper to be a unit 119889-dimensional torus R119889
Z119889 witha flat metric 120588 and the volume V We also associate with119894 isin Γ a bosonic Fock-Hilbert space H(119894) ≃ H Here H =
oplus119896=01
H119896whereH
119896= Lsym
2(119872
119896) is the subspace in L
2(119872
119896)
formed by functions symmetric under a permutation of thevariables Given a finite set Λ sub Γ we setH(Λ) = otimes
119894isinΛH(119894)
An element 120601 isin H(Λ) is a complex function
xlowastΛisin 119872
lowastΛ997891997888rarr 120601 (xlowast
Λ) (5)
Here xlowastΛis a collection xlowast(119895) 119895 isin Λ of finite point sets
xlowast(119895) sub 119872 associated with sites 119895 isin Λ Following [1] we callxlowast(119895) a particle configuration at site 119895 (which can be empty)and xlowast
Λa particle configuration in or over Λ The space
119872
lowastΛ of particle configurations inΛ can be represented as theCartesian product (119872lowast
)
timesΛ where 119872
lowast is the disjoint union⋃
119896=01119872
(119896) and 119872
(119896) is the collection of (unordered) 119896-point subsets of119872 (One can consider119872(119896) as the factor of theldquooff-diagonalrdquo set 119872119896
=in the Cartesian power 119872119896 under the
equivalence relation induced by the permutation group oforder 119896) The norm and the scalar product in H
Λare given
by
1003817100381710038171003817
1206011003817100381710038171003817
= (int
119872lowastΛ
1003816100381610038161003816
120601(xlowastΛ)
1003816100381610038161003816
2dxlowastΛ)
12
⟨1206011120601
2⟩ = int
119872lowastΛ
1206011(xlowast
Λ)120601
2(xlowast
Λ)dxlowast
Λ
(6)
where measure dxlowastΛis the product times
119895isinΛdxlowast(119895) and dxlowast(119895) is
the Poissonian sum measure on119872
lowast
dxlowast (119895) = sum
119896=01
1 (♯xlowast (119895) = 119896)
1
119896
prod
119909isinxlowast(119895)dV (119909) 119890minusV(119872)
(7)
Here V(119872) is the volume of torus119872As in [1] we assume that an action
(g 119909) isin G times119872 997891997888rarr g119909 isin 119872 (8)
is given of a group G that is a Euclidean space or a torus ofdimension 119889
1015840le 119889 The action is written as
g119909 = 119909 + 120579119860 mod 1 (9)
Here vector 120579 = (1205791 120579
1198891015840) with components 120579
119897isin [0 1)
and 120579119860 is the 119889-dimensional vector 120579 = ((120579119860)1 (120579119860)
119889)
representing the element g where 119860 is a (1198891015840 times 119889) matrix ofcolumn rank 1198891015840 with rational entries The action of G is liftedto unitary operators U
Λ(g) inH
Λ
UΛ(g)120601 (xlowast
Λ) = 120601 (g
minus1xlowastΛ) (10)
where gminus1xlowastΛ= gminus1xlowast(119895) 119895 isin Λ and gminus1xlowast(119895) = gminus1119909 119909 isin
xlowast(119895)The generally accepted view is that the Hubbard model
is a highly oversimplified model for strongly interactingelectrons in a solidTheHubbardmodel is a kind ofminimummodel which takes into account quantummechanicalmotionof electrons in a solid and nonlinear repulsive interactionbetween electrons There is little doubt that the model is toosimple to describe actual solids faithfully [5] In our contextthe Hubbard Hamiltonian H
Λof the system in Λ acts as
follows
(HΛ120601) (xlowast
Λ) =
[
[
minus
1
2
sum
119895isinΛ
sum
119909isinxlowast(119895)Δ
(119909)
119895+ sum
119895isinΛ
sum
119909isinxlowast(119895)119880
(1)(119909)
+
1
2
sum
119895isinΛ
sum
1199091199091015840isinxlowast(119895)
1 (119909 = 119909
1015840)119880
(2)(119909 119909
1015840)
+
1
2
sum
1198951198951015840isinΛ
1 (119895 = 119895
1015840) 119869 (d (119895 119895
1015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)]
]
120601 (xlowastΛ)
+ sum
1198951198951015840isinΛ
12058211989511989510158401 (♯xlowast (119895) ge 1 ♯xlowast (1198951015840) lt 120581)
times sum
119909isinxlowast(119895)int
119872
V (d119910)
times [120601 (xlowast(119895119909)rarr (1198951015840
119910)
Λ) minus 120601 (xlowast
Λ)]
(11)
Advances in Mathematical Physics 3
HereΔ(119909)
119895means the Laplacian in variable119909 isin xlowast(119895) Next ♯xlowast
stands for the cardinality of the particle configuration xlowast (ie♯xlowast = 119896 when xlowast isin 119872
(119896)) and the parameter 120581 is introducedin (17) (Symbol ♯ will be used for denoting the cardinalityof a general (finite) set for example ♯Λ means the numberof vertices in Λ) Further xlowast(119895119909)rarr (119895
1015840
119910)
Λdenotes the particle
configuration with the point 119909 isin xlowast(119895) removed and point 119910added to xlowast(1198951015840)
As in [1] we also consider a Hamiltonian HΛ|xlowastΓΛ
in anexternal field generated by a configuration xlowast
ΓΛ= xlowast(1198951015840)
119895
1015840isin Γ Λ isin 119872
lowastΓΛ where Γ sube Γ is a (finite or infinite)collection of vertices More precisely we only consider xlowast
ΓΛ
with ♯xlowast(1198951015840) le 120581 (see (17) below) and set
(HΛ|xlowastΓΛ
120601) (xlowastΛ)
=[
[
minus
1
2
sum
119895isinΛ
sum
119909isinxlowast(119895)Δ
(119909)
119895+ sum
119895isinΛ
sum
119909isinxlowast(119895)119880
(1)(119909)
+
1
2
sum
119895isinΛ
sum
1199091199091015840isinxlowast(119895)
1 (119909 = 119909
1015840)119880
(2)(119909 119909
1015840)
+
1
2
sum
1198951198951015840isinΛ
1 (119895 = 119895
1015840) 119869 (d (119895 119895
1015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)]
]
120601 (xlowastΛ)
+ sum
119895isinΛ1198951015840isinΓΛ
119869 (d (119895 1198951015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)120601 (xlowast
Λ)
+ sum
1198951198951015840isinΛ
12058211989511989510158401 (♯xlowast (119895) ge 1 ♯xlowast (1198951015840) lt 120581)
times sum
119909isinxlowast(119895)
int
119872
V (d119910) [120601 (xlowast(119895119909)rarr (1198951015840
119910)
Λ) minus 120601 (xlowast
Λ)]
(12)
The novel elements in (11) and (12) compared with [1]are the presence of on-site potentials 119880(1) and 119880
(2) and thesummand involving transition rates 120582
1198951198951015840 ge 0 for jumps of a
particle from site 119895 to 1198951015840We will suppose that 120582
1198951198951015840 vanishes if the graph distance
d(119895 1198951015840) gt 1 We will also assume uniform boundedness
sup [1205821198951198951015840 (119909119872) 119895 119895
1015840isin Γ 119909 isin 119872] lt infin (13)
in view of (1) it implies that the total exit ratesum
1198951015840d(1198951198951015840)=1 120582119895119895
1015840(119909119872) from site 119895 is uniformly boundedThese conditions are not sharp and can be liberalized
The model under consideration can be considered as ageneralization of the Hubbard model [6] (in its bosonic ver-sion) Its mathematical justification includes the following(a) An opportunity to introduce a Fock space formalismincorporates a number of new features For instance afermonic version of the model (not considered here) emergesnaturally when the bosonic Fock space H(119894) is replaced bya fermonic one Another opening provided by this modelis a possibility to consider random potentials 119880(1) 119880(2) and119881 which would yield a sound generalization of the Mott-Andersonmodel (b) Introducing jumpsmakes a step towardsa treatment of a model of a quantum (Bose-) gas whereparticles ldquoliverdquo in a single Fock space For example a systemof interacting quantum particles is originally confined to aldquoboxrdquo in a Euclidean space with or without ldquointernalrdquo degreesof freedom In the thermodynamical limit the box expandsto the whole Euclidean space In a two-dimensional modelof a quantum gas one expects a phenomenon of invarianceunder space-translations one hopes to be able to address thisissue in future publications (c) A model with jumps can beanalysed by means of the theory of Markov processes whichprovides a developed methodology
Physically speaking the model with jumps covers a situ-ation where ldquolightrdquo quantum particles are subject to a ldquoran-domrdquo force and change their ldquolocationrdquo This class of modelsis interesting from the point of view of transport phenomenathat theymay display (An analogywith the famousAndersonmodel in its multiparticle version inevitably comes tomind cf eg [7]) Methodologically such systems occupyan ldquointermediaterdquo place between models where quantumparticles are ldquofixedrdquo forever to their designated locations (asin [1]) andmodels where quantumparticlesmove in the samespace (a Bose-gas considered in [8 9]) In particular thiswork provides a bridge between [1 8 9] reading this paperahead of [8 9] might help an interested reader to get through[8 9] at a much quicker pace
We would like to note an interesting problem of analysisof the small-mass limit (cf [10]) from the point of Mermin-Wagner phenomena
13 Assumptions on the Potentials The between-sites poten-tial 119881 is assumed to be of class 1198622 Consequently 119881 and itsfirst and second derivatives satisfy uniform bounds Namelyforall119909
1015840 119909
10158401015840isin 119872
minus119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
nablax119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
10038161003816100381610038161003816
nablaxx1015840119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
le 119881 (14)
Here 119909 and 1199091015840 run through the pairs of variables 119909 1199091015840 A simi-lar property is assumed for the on-site potential119880(1) (here weneed only a 1198621 smoothness)
minus119880
(1)(119909)
10038161003816100381610038161003816
nablax119880(1)
(119909)
10038161003816100381610038161003816
le 119880
(1)
119909 isin 119872 (15)
Note that for119881 and119880(1) the bounds are imposed on their neg-ative parts only
As to 119880(2) we suppose that (a)
119880
(2)(119909 119909
1015840) = +infin when 1003816
1003816100381610038161003816
119909 minus 119909
101584010038161003816100381610038161003816
le 120588 (16)
4 Advances in Mathematical Physics
and (b) exist a 1198621-function (119909 119909
1015840) 997891rarr
119880
(2)(119909 119909
1015840) isin R such that
119880
(2)(119909 119909
1015840) =
119880
(2)(119909 119909
1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)
stands for the (flat) Riemannian distance between points119909 119909
1015840isin 119872 As a result of (16) there exists a ldquohard corerdquo of
diameter 120588 and a given atom cannot ldquoholdrdquo more than
120581 = lceil
V (119872)
V (119861 (120588))
rceil (17)
particles where V(119861(120588)) is the volume of a 119889-dimensional ballof diameter 120588 We will also use the bound
minus119880
(2)(119909 119909
1015840)
10038161003816100381610038161003816
nablax119880(2)
(119909 119909
1015840)
10038161003816100381610038161003816
le 119880
(2)
119909 119909
1015840isin 119872 (18)
Formally (16) means that the operators in (11) and (12) areconsidered for functions120601(xlowast
Λ) vanishingwhen in the particle
configuration xlowastΛ= xlowast(119895) 119895 isin Λ the cardinality ♯xlowast(119895) gt 120581
for some 119895 isin ΛThe function 119869 119903 isin (0infin) 997891rarr 119869(119903) ge 0 is assumed
monotonically nonincreasing with 119903 and obeying the relation119869(119897) rarr 0 as 119897 rarr infin where
119869 (119897) = sup[
[
sum
11989510158401015840isinΓ
119869 (d (1198951015840 119895
10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895
1015840isin Γ
]
]
lt infin
(19)
Additionally let 119869(119903) be such that
119869
lowast= sup[
[
sum
1198951015840isinΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
119895 isin Γ]
]
lt infin (20)
Next we assume that the functions 119880(1) 119880(2) and 119881 are g-invariant forall119909 1199091015840 isin 119872 and g isin G
119880
(1)(119909) = 119880
(1)(g119909)
119880
(2)(119909 119909
1015840) = 119880
(2)(g119909 g119909
1015840)
119881 (119909 119909
1015840) = 119881 (g119909 g119909
1015840)
(21)
In the following we will need to bound the fugacity (oractivity cf (25)) 119911 in terms of the other parameters of themodel
119911119890
Θlt 1 where Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881) (22)
14 The Gibbs State in a Finite Volume Define the particlenumber operator N
Λ with the action
NΛ120601 (xlowast
Λ) = ♯xlowast
Λ120601 (xlowast
Λ) xlowast
Λisin 119872
lowastΛ (23)
Here for a given xlowastΛ= xlowast(119895) 119895 isin Λ ♯xlowast
Λstands for the total
number of particles in configuration xlowastΛ
♯xlowastΛ= sum
119895isinΛ
♯xlowast (119895) (24)
The standard canonical variable associated withNΛis activity
119911 isin (0infin)The Hamiltonians (11) and (12) are self-adjoint (on the
natural domains) in H(Λ) Moreover they are positivedefinite and have a discrete spectrum cf [14] Furthermoreforall119911 120573 gt 0 H
Λand H
Λ|xlowastΓΛ
give rise to positive-definite trace-class operators G
Λ= G
119911120573Λand G
Λ|xlowastΓΛ
= G119911120573Λ|xlowast
ΓΛ
GΛ= 119911
NΛ exp [minus120573H
Λ]
GΛ|xlowastΓΛ
= 119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
]
(25)
We would like to stress that the full range of variables 119911 120573 gt 0
is allowedhere because of the hard-core condition (16) it doesnot allow more than 120581♯Λ particles in Λ where ♯Λ stands forthe number of vertices in Λ However while passing to thethermodynamic limit we will need to control 119911 and 120573
Definition 1 We will call GΛand G
Λ|xlowastΓΛ
the Gibbs operatorsin volume Λ for given values of 119911 and 120573 (andmdashin the case ofG
Λ|xlowastΓΛ
mdashwith the boundary condition xlowastΓΛ
)The Gibbs operators in turn give rise to the Gibbs states120593Λ= 120593
120573119911Λand 120593
Λ|xΓΛ
= 120593120573119911Λ|x
ΓΛ
at temperature 120573minus1 andactivity 119911 in volume Λ These are linear positive normalizedfunctionals on the 119862
lowast-algebra BΛof bounded operators in
spaceHΛ
120593Λ(A) = trH
Λ
(RΛA)
120593Λ|xΓΛ
(A) = trHΛ
(RΛ|xΓΛ
A) A isin BΛ
(26)
where
RΛ=
GΛ
Ξ (Λ) with Ξ (Λ) = Ξ
119911120573(Λ) = trH
Λ
GΛ (27)
RΛ|xlowastΓΛ
=
GΛ|xΓΛ
Ξ (Λ | xlowastΓΛ
)
with Ξ (Λ | xlowastΓΛ
) = Ξ119911120573
(Λ | xlowastΓΛ
)
= trHΛ
(119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
])
(28)
The hard-core assumption (16) yields that the quantitiesΞ(Λ) and Ξ
119911120573(Λ | xlowast
ΓΛ) are finite formally these facts will
be verified by virtue of the Feynman-Kac representation
Definition 2 Whenever Λ0sub Λ the 119862lowast-algebraB
Λ0 is iden-
tified with the 119862lowast subalgebra inBΛformed by the operators
of the form A0otimes I
ΛΛ0 Consequently the restriction 120593Λ
0
Λof
state 120593Λto 119862lowast-algebraB
Λ0 is given by
120593Λ0
Λ(A
0) = trH
Λ0
(RΛ0
ΛA0) A
0isin B
Λ0 (29)
where
RΛ0
Λ= trH
ΛΛ0
RΛ (30)
Advances in Mathematical Physics 5
Operators RΛ0
Λ(we again call them RDMs) are positive defi-
nite and have trHΛ0
RΛ0
Λ= 1 They also satisfy the compatibil-
ity property forallΛ0sub Λ
1sub Λ
RΛ0
Λ= trH
Λ1Λ0
RΛ1
Λ (31)
In a similar fashion one defines functionals 120593Λ0
Λ|xlowastΓΛ
and oper-
ators RΛ0
Λ|xlowastΓΛ
with the same properties
15 Limiting Gibbs States The concept of a limiting Gibbsstate is related to notion of a quasilocal 119862lowast-algebra see [14]For the class of systems under consideration the constructionof the quasilocal 119862lowast-algebraB
Γis done along the same lines
as in [1]BΓis the norm completion of the 119862lowast algebra (B0
Γ) =
lim ind119899rarrinfin
BΛ119899
Any family of positive-definite operatorsRΛ0
in spacesHΛ0 of trace one whereΛ0 runs through finite
subsets of Γ with the compatibility property
RΛ1
= trHΛ0Λ1
RΛ0
Λ
1sub Λ
0 (32)
determines a state ofBΓ see [12 13]
Finally we introduce unitary operators UΛ0(g) g isin G in
HΛ0
UΛ120601 (xlowast
Λ0) = 120601 (g
minus1xlowastΛ0) (33)
where
gminus1xlowast
Λ0 = g
minus1xlowast (119895) 119895 isin Λ
0
gminus1xlowast (119895) = g
minus1119909 119909 isin xlowast (119895)
(34)
Theorem3 Assuming the conditions listed above for all 119911 120573 isin
(0 +infin) satisfying (22) and a finite Λ
0sub Γ operators RΛ
0
Λ
form a compact sequence in the trace-norm topology in HΛ0
as Λ Γ Furthermore given any family of (finite or infinite)sets Γ = Γ(Λ) sube Γ and configurations xlowast
ΓΛ= xlowast(119894) 119894 isin Γ Λ
with ♯xlowast(119894) lt 120581 operatorsRΛ0
Λ|xΓΛ
also forma compact sequence
in the trace-norm topology Any limit point RΛ0
for RΛ0
Λ or
RΛ0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operator in H(Λ
0)
of trace one Moreover if Λ1sub Λ
0 and RΛ0
and RΛ1
are thelimits forRΛ
0
ΛandRΛ
1
Λor forRΛ
0
Λ|xlowastΓΛ
andRΛ1
Λ|xlowastΓΛ
along the samesubsequence Λ
119904 Γ then the property (32) holds true
Consequently theGibbs states120593Λand120593
Λ|xlowastΓΛ
form compactsequences as Λ Γ
Remark 4 In fact the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (ΓE)only under an assumption that the degree of the vertices inΓ is uniformy bounded
Definition 5 Any limit point 120593 for states 120593Λand 120593
Λ|xlowastΓΛ
iscalled a limiting Gibbs state (for given 119911 120573 isin (0 +infin))
Theorem 6 Under the condition (22) any limiting pointRΛ0
for RΛ
0
Λ or RΛ
0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operatorof trace one commuting with operators U
Λ0(g) forallg isin G
UΛ0(g)
minus1RΛ0
UΛ0 (g) = RΛ
0
(35)
Accordingly any limiting Gibbs state 120593 of B determined by afamily of limiting operators RΛ
0
obeying (35) satisfies the cor-responding invariance property forall finite Λ0
sub Γ any A isin BΛ0
and g isin G
120593 (A) = 120593 (UΛ0(g)
minus1AUΛ0 (g)) (36)
Remarks (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (ie absence of phase transi-tions)
(2) Properties (35) and (36) are trivially fulfilled for thelimiting points RΛ
0
and 120593 of families RΛ0
Λ and 120593
Λ How-
ever they require a proof for the limit points of the familiesRΛ0
Λ|xlowastΓΛ
and 120593Λ|xlowastΓΛ
The set of limiting Gibbs states (which is nonempty due
to Theorem 3) is denoted by G0 In Section 3 we describe aclassG sup G0 of states of119862lowast-algebraB satisfying the FK-DLRequation similar to that in [1]
2 Feynman-Kac Representations forthe RDM Kernels in a Finite Volume
21 The Representation for the Kernels of the Gibbs OperatorsA starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels K
Λ(xlowast
Λ ylowast
Λ) =
K120573119911Λ
(xlowastΛ ylowast
Λ) and F
Λ(xlowast
Λ ylowast
Λ) = F
120573119911Λ(xlowast
Λ ylowast
Λ) of operators
GΛand R
Λ
Definition 7 Given (119909 119894) (119910 119895) isin 119872 times Γ 119882120573
(119909119894)(119910119895)denotes
the space of path or trajectories 120596 = 120596(119909119894)(119910119895)
in 119872 times Γ oftime-length 120573 with the end-points (119909 119894) and (119910 119895) Formally120596 isin 119882
120573
(119909119894)(119910119895)is defined as follows
120596 120591 isin [0 120573] 997891997888rarr 120596 (120591) = (119906 (120596 120591) 119897 (120596 120591)) isin 119872 times Γ
120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)
120596 has finitely many jumps on [0 120573]
if a jump occurs at time 120591 then d [119897 (120596 120591minus) 119897 (120596 120591)]=1
(37)
The notation 120596(120591) and its alternative (119906(120596 120591) 119897(120596 120591)) forthe position and the index of trajectory 120596 at time 120591 will beemployed as equal in rights We use the term the temporalsection (or simply the section) of path 120596 at time 120591
Definition 8 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ A matching (or pairing) 120574 between xlowast
Λand ylowast
Λis
6 Advances in Mathematical Physics
defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ
119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ
and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast
Λ= ♯ylowast
Λ these properties are equivalent) It is convenient
to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]
Next consider the Cartesian product
119882
120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
120573
(119909119894)120574(119909119894) (38)
and the disjoint union
119882
120573
xlowastΛylowastΛ
= ⋃
120574
119882
120573
xlowastΛylowastΛ120574 (39)
Accordingly an element 120596Λ
isin 119882
120573
xlowastΛylowastΛ120574in (38) represents a
collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573
starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a
path configuration in (or over) Λ
The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation
We will work with standard sigma algebras (generated bycylinder sets) in119882
120573
(119909119894)(119910119895)119882120573
xlowastΛylowastΛ120574 and119882
120573
xlowastΛylowastΛ
Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times
Λ 997891rarr 120595(119909 119894) by
L120595 (119909 119894) = minus
1
2
Δ120595 (119909 119894)
+ sum
119895d(119894119895)=1
120582119894119895int
119872
V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]
(40)
In the probabilistic literature such processes are referred toas Levy processes see for example [14]
Pictorially a trajectory of process 120585 moves along 119872
according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum
119895d(119894119895)=1 120582119894119895 In other words while following a Brownian
motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582
119894119895V(d119910)
After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on
For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573
(119909119894)(119910119895)the nonnormalised measure on 119882
120573
(119909119894)(119910119895)induced
by 120585 That is under measure P120573
(119909119894)(119910119895)the trajectory at time
120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)
= P120573
(119909119894)(119910119895)(119882
120573
(119909119894)(119910119895)) is given by
119901(119909119894)(119910119895)
= 1 (119894 = 119895) 119901
120573
119872(119909 119910) exp[
[
minus120573 sum
119895d(119894119895)=1
120582119894119895]
]
+ sum
119896ge1
sum
1198970=1198941198971119897119896119897119896+1
=119895
prod
0le119904le119896
1 (d (119897119904 119897119904+1
) = 1)
times 120582119897119904119897119904+1
int
120573
0
d120591119904exp[
[
minus (120591119904+1
minus 120591119904) sum
119895d(119897119904119895)=1
120582119897119904119895]
]
times 1 (0 = 1205910lt 120591
1lt sdot sdot sdot lt 120591
119896lt 120591
119896+1= 120573)
(41)
where 119901120573
119872(119909 119910) denotes the transition probability density for
the Brownian motion to pass from 119909 to 119910 on119872 in time 120573
119901
120573
119872(119909 119910) =
1
(2120587120573)
1198892
times sum
119899=(1198991119899119889)isinZ119889
exp(minus
1003816100381610038161003816
119909 minus 119910 + 119899
1003816100381610038161003816
2
2120573
)
(42)
In view of (13) the quantity 119901(119909119894)(119910119895)
and its derivatives areuniformly bounded
119901(119909119894)(119910119895)
10038161003816100381610038161003816
nabla119909119901(119909119894)(119910119895)
10038161003816100381610038161003816
10038161003816100381610038161003816
nabla119910119901(119909119894)(119910119895)
10038161003816100381610038161003816
le 119901119872
119909 119910 isin 119872 119894 119895 isin Γ
(43)
where 119901119872
= 119901119872(120573) isin (0 +infin) is a constant
We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future
Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations overΛ with ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a pairing between xlowast
Λand ylowast
Λ
Then Plowast
xlowastΛylowastΛ120574denotes the product measure on119882
120573
xlowastΛylowastΛ120574
P120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)P120573
(119909119894)120574(119909119894) (44)
Furthermore P120573
xlowastΛylowastΛ
stands for the sum measure on119882
120573
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
= sum
120574
P120573
xlowastΛylowastΛ120574 (45)
According to Definition 10 under the measure P120573
xlowastΛylowastΛ120574
the trajectories 120596119909119894
isin 119882
120573
(119909119894)120574(119909119894)constituting 120596
Λare inde-
pendent components (Here the term independence is usedin the measure-theoretical sense)
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
The bi-dimensionality property is expressed in the bound
0 lt sup [1119899
♯Σ (119895 119899) 119895 isin Γ 119899 = 1 2 ] lt infin (2)
where Σ(119895 119899) stands for the set of vertices in Γ at the graphdistance 119899 from 119895 isin Γ (a sphere of radius 119899 around 119895)
Σ (119895 119899) = 119895
1015840isin Γ d (119895 119895
1015840) = 119899 (3)
(The graph distance d(119895 1198951015840) = dΓE(119895 119895
1015840) between 119895 119895
1015840isin Γ is
determined as theminimal length of a path on (ΓE) joining 119895and 1198951015840)This implies that for any 119900 isin Γ the cardinality ♯Λ(119900 119899)of the ball
Λ (119900 119899) = 119895
1015840isin Γ d (119900 119895
1015840) le 119899 (4)
grows at most quadratically with 119899A justification for putting a quantum system on a graph
can be that graph-like structures become increasingly popu-lar in rigorous StatisticalMechanics for example in quantumgravity Namely see [2ndash4] On the other hand a number ofproperties of Gibbs ensembles do not depend upon ldquoregular-ityrdquo of an underlying spatial geometry
12 A BosonicModel in the Fock Space With each vertex 119894 isin Γ
we associate a copy of a compact manifold119872 which we takein this paper to be a unit 119889-dimensional torus R119889
Z119889 witha flat metric 120588 and the volume V We also associate with119894 isin Γ a bosonic Fock-Hilbert space H(119894) ≃ H Here H =
oplus119896=01
H119896whereH
119896= Lsym
2(119872
119896) is the subspace in L
2(119872
119896)
formed by functions symmetric under a permutation of thevariables Given a finite set Λ sub Γ we setH(Λ) = otimes
119894isinΛH(119894)
An element 120601 isin H(Λ) is a complex function
xlowastΛisin 119872
lowastΛ997891997888rarr 120601 (xlowast
Λ) (5)
Here xlowastΛis a collection xlowast(119895) 119895 isin Λ of finite point sets
xlowast(119895) sub 119872 associated with sites 119895 isin Λ Following [1] we callxlowast(119895) a particle configuration at site 119895 (which can be empty)and xlowast
Λa particle configuration in or over Λ The space
119872
lowastΛ of particle configurations inΛ can be represented as theCartesian product (119872lowast
)
timesΛ where 119872
lowast is the disjoint union⋃
119896=01119872
(119896) and 119872
(119896) is the collection of (unordered) 119896-point subsets of119872 (One can consider119872(119896) as the factor of theldquooff-diagonalrdquo set 119872119896
=in the Cartesian power 119872119896 under the
equivalence relation induced by the permutation group oforder 119896) The norm and the scalar product in H
Λare given
by
1003817100381710038171003817
1206011003817100381710038171003817
= (int
119872lowastΛ
1003816100381610038161003816
120601(xlowastΛ)
1003816100381610038161003816
2dxlowastΛ)
12
⟨1206011120601
2⟩ = int
119872lowastΛ
1206011(xlowast
Λ)120601
2(xlowast
Λ)dxlowast
Λ
(6)
where measure dxlowastΛis the product times
119895isinΛdxlowast(119895) and dxlowast(119895) is
the Poissonian sum measure on119872
lowast
dxlowast (119895) = sum
119896=01
1 (♯xlowast (119895) = 119896)
1
119896
prod
119909isinxlowast(119895)dV (119909) 119890minusV(119872)
(7)
Here V(119872) is the volume of torus119872As in [1] we assume that an action
(g 119909) isin G times119872 997891997888rarr g119909 isin 119872 (8)
is given of a group G that is a Euclidean space or a torus ofdimension 119889
1015840le 119889 The action is written as
g119909 = 119909 + 120579119860 mod 1 (9)
Here vector 120579 = (1205791 120579
1198891015840) with components 120579
119897isin [0 1)
and 120579119860 is the 119889-dimensional vector 120579 = ((120579119860)1 (120579119860)
119889)
representing the element g where 119860 is a (1198891015840 times 119889) matrix ofcolumn rank 1198891015840 with rational entries The action of G is liftedto unitary operators U
Λ(g) inH
Λ
UΛ(g)120601 (xlowast
Λ) = 120601 (g
minus1xlowastΛ) (10)
where gminus1xlowastΛ= gminus1xlowast(119895) 119895 isin Λ and gminus1xlowast(119895) = gminus1119909 119909 isin
xlowast(119895)The generally accepted view is that the Hubbard model
is a highly oversimplified model for strongly interactingelectrons in a solidTheHubbardmodel is a kind ofminimummodel which takes into account quantummechanicalmotionof electrons in a solid and nonlinear repulsive interactionbetween electrons There is little doubt that the model is toosimple to describe actual solids faithfully [5] In our contextthe Hubbard Hamiltonian H
Λof the system in Λ acts as
follows
(HΛ120601) (xlowast
Λ) =
[
[
minus
1
2
sum
119895isinΛ
sum
119909isinxlowast(119895)Δ
(119909)
119895+ sum
119895isinΛ
sum
119909isinxlowast(119895)119880
(1)(119909)
+
1
2
sum
119895isinΛ
sum
1199091199091015840isinxlowast(119895)
1 (119909 = 119909
1015840)119880
(2)(119909 119909
1015840)
+
1
2
sum
1198951198951015840isinΛ
1 (119895 = 119895
1015840) 119869 (d (119895 119895
1015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)]
]
120601 (xlowastΛ)
+ sum
1198951198951015840isinΛ
12058211989511989510158401 (♯xlowast (119895) ge 1 ♯xlowast (1198951015840) lt 120581)
times sum
119909isinxlowast(119895)int
119872
V (d119910)
times [120601 (xlowast(119895119909)rarr (1198951015840
119910)
Λ) minus 120601 (xlowast
Λ)]
(11)
Advances in Mathematical Physics 3
HereΔ(119909)
119895means the Laplacian in variable119909 isin xlowast(119895) Next ♯xlowast
stands for the cardinality of the particle configuration xlowast (ie♯xlowast = 119896 when xlowast isin 119872
(119896)) and the parameter 120581 is introducedin (17) (Symbol ♯ will be used for denoting the cardinalityof a general (finite) set for example ♯Λ means the numberof vertices in Λ) Further xlowast(119895119909)rarr (119895
1015840
119910)
Λdenotes the particle
configuration with the point 119909 isin xlowast(119895) removed and point 119910added to xlowast(1198951015840)
As in [1] we also consider a Hamiltonian HΛ|xlowastΓΛ
in anexternal field generated by a configuration xlowast
ΓΛ= xlowast(1198951015840)
119895
1015840isin Γ Λ isin 119872
lowastΓΛ where Γ sube Γ is a (finite or infinite)collection of vertices More precisely we only consider xlowast
ΓΛ
with ♯xlowast(1198951015840) le 120581 (see (17) below) and set
(HΛ|xlowastΓΛ
120601) (xlowastΛ)
=[
[
minus
1
2
sum
119895isinΛ
sum
119909isinxlowast(119895)Δ
(119909)
119895+ sum
119895isinΛ
sum
119909isinxlowast(119895)119880
(1)(119909)
+
1
2
sum
119895isinΛ
sum
1199091199091015840isinxlowast(119895)
1 (119909 = 119909
1015840)119880
(2)(119909 119909
1015840)
+
1
2
sum
1198951198951015840isinΛ
1 (119895 = 119895
1015840) 119869 (d (119895 119895
1015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)]
]
120601 (xlowastΛ)
+ sum
119895isinΛ1198951015840isinΓΛ
119869 (d (119895 1198951015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)120601 (xlowast
Λ)
+ sum
1198951198951015840isinΛ
12058211989511989510158401 (♯xlowast (119895) ge 1 ♯xlowast (1198951015840) lt 120581)
times sum
119909isinxlowast(119895)
int
119872
V (d119910) [120601 (xlowast(119895119909)rarr (1198951015840
119910)
Λ) minus 120601 (xlowast
Λ)]
(12)
The novel elements in (11) and (12) compared with [1]are the presence of on-site potentials 119880(1) and 119880
(2) and thesummand involving transition rates 120582
1198951198951015840 ge 0 for jumps of a
particle from site 119895 to 1198951015840We will suppose that 120582
1198951198951015840 vanishes if the graph distance
d(119895 1198951015840) gt 1 We will also assume uniform boundedness
sup [1205821198951198951015840 (119909119872) 119895 119895
1015840isin Γ 119909 isin 119872] lt infin (13)
in view of (1) it implies that the total exit ratesum
1198951015840d(1198951198951015840)=1 120582119895119895
1015840(119909119872) from site 119895 is uniformly boundedThese conditions are not sharp and can be liberalized
The model under consideration can be considered as ageneralization of the Hubbard model [6] (in its bosonic ver-sion) Its mathematical justification includes the following(a) An opportunity to introduce a Fock space formalismincorporates a number of new features For instance afermonic version of the model (not considered here) emergesnaturally when the bosonic Fock space H(119894) is replaced bya fermonic one Another opening provided by this modelis a possibility to consider random potentials 119880(1) 119880(2) and119881 which would yield a sound generalization of the Mott-Andersonmodel (b) Introducing jumpsmakes a step towardsa treatment of a model of a quantum (Bose-) gas whereparticles ldquoliverdquo in a single Fock space For example a systemof interacting quantum particles is originally confined to aldquoboxrdquo in a Euclidean space with or without ldquointernalrdquo degreesof freedom In the thermodynamical limit the box expandsto the whole Euclidean space In a two-dimensional modelof a quantum gas one expects a phenomenon of invarianceunder space-translations one hopes to be able to address thisissue in future publications (c) A model with jumps can beanalysed by means of the theory of Markov processes whichprovides a developed methodology
Physically speaking the model with jumps covers a situ-ation where ldquolightrdquo quantum particles are subject to a ldquoran-domrdquo force and change their ldquolocationrdquo This class of modelsis interesting from the point of view of transport phenomenathat theymay display (An analogywith the famousAndersonmodel in its multiparticle version inevitably comes tomind cf eg [7]) Methodologically such systems occupyan ldquointermediaterdquo place between models where quantumparticles are ldquofixedrdquo forever to their designated locations (asin [1]) andmodels where quantumparticlesmove in the samespace (a Bose-gas considered in [8 9]) In particular thiswork provides a bridge between [1 8 9] reading this paperahead of [8 9] might help an interested reader to get through[8 9] at a much quicker pace
We would like to note an interesting problem of analysisof the small-mass limit (cf [10]) from the point of Mermin-Wagner phenomena
13 Assumptions on the Potentials The between-sites poten-tial 119881 is assumed to be of class 1198622 Consequently 119881 and itsfirst and second derivatives satisfy uniform bounds Namelyforall119909
1015840 119909
10158401015840isin 119872
minus119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
nablax119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
10038161003816100381610038161003816
nablaxx1015840119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
le 119881 (14)
Here 119909 and 1199091015840 run through the pairs of variables 119909 1199091015840 A simi-lar property is assumed for the on-site potential119880(1) (here weneed only a 1198621 smoothness)
minus119880
(1)(119909)
10038161003816100381610038161003816
nablax119880(1)
(119909)
10038161003816100381610038161003816
le 119880
(1)
119909 isin 119872 (15)
Note that for119881 and119880(1) the bounds are imposed on their neg-ative parts only
As to 119880(2) we suppose that (a)
119880
(2)(119909 119909
1015840) = +infin when 1003816
1003816100381610038161003816
119909 minus 119909
101584010038161003816100381610038161003816
le 120588 (16)
4 Advances in Mathematical Physics
and (b) exist a 1198621-function (119909 119909
1015840) 997891rarr
119880
(2)(119909 119909
1015840) isin R such that
119880
(2)(119909 119909
1015840) =
119880
(2)(119909 119909
1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)
stands for the (flat) Riemannian distance between points119909 119909
1015840isin 119872 As a result of (16) there exists a ldquohard corerdquo of
diameter 120588 and a given atom cannot ldquoholdrdquo more than
120581 = lceil
V (119872)
V (119861 (120588))
rceil (17)
particles where V(119861(120588)) is the volume of a 119889-dimensional ballof diameter 120588 We will also use the bound
minus119880
(2)(119909 119909
1015840)
10038161003816100381610038161003816
nablax119880(2)
(119909 119909
1015840)
10038161003816100381610038161003816
le 119880
(2)
119909 119909
1015840isin 119872 (18)
Formally (16) means that the operators in (11) and (12) areconsidered for functions120601(xlowast
Λ) vanishingwhen in the particle
configuration xlowastΛ= xlowast(119895) 119895 isin Λ the cardinality ♯xlowast(119895) gt 120581
for some 119895 isin ΛThe function 119869 119903 isin (0infin) 997891rarr 119869(119903) ge 0 is assumed
monotonically nonincreasing with 119903 and obeying the relation119869(119897) rarr 0 as 119897 rarr infin where
119869 (119897) = sup[
[
sum
11989510158401015840isinΓ
119869 (d (1198951015840 119895
10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895
1015840isin Γ
]
]
lt infin
(19)
Additionally let 119869(119903) be such that
119869
lowast= sup[
[
sum
1198951015840isinΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
119895 isin Γ]
]
lt infin (20)
Next we assume that the functions 119880(1) 119880(2) and 119881 are g-invariant forall119909 1199091015840 isin 119872 and g isin G
119880
(1)(119909) = 119880
(1)(g119909)
119880
(2)(119909 119909
1015840) = 119880
(2)(g119909 g119909
1015840)
119881 (119909 119909
1015840) = 119881 (g119909 g119909
1015840)
(21)
In the following we will need to bound the fugacity (oractivity cf (25)) 119911 in terms of the other parameters of themodel
119911119890
Θlt 1 where Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881) (22)
14 The Gibbs State in a Finite Volume Define the particlenumber operator N
Λ with the action
NΛ120601 (xlowast
Λ) = ♯xlowast
Λ120601 (xlowast
Λ) xlowast
Λisin 119872
lowastΛ (23)
Here for a given xlowastΛ= xlowast(119895) 119895 isin Λ ♯xlowast
Λstands for the total
number of particles in configuration xlowastΛ
♯xlowastΛ= sum
119895isinΛ
♯xlowast (119895) (24)
The standard canonical variable associated withNΛis activity
119911 isin (0infin)The Hamiltonians (11) and (12) are self-adjoint (on the
natural domains) in H(Λ) Moreover they are positivedefinite and have a discrete spectrum cf [14] Furthermoreforall119911 120573 gt 0 H
Λand H
Λ|xlowastΓΛ
give rise to positive-definite trace-class operators G
Λ= G
119911120573Λand G
Λ|xlowastΓΛ
= G119911120573Λ|xlowast
ΓΛ
GΛ= 119911
NΛ exp [minus120573H
Λ]
GΛ|xlowastΓΛ
= 119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
]
(25)
We would like to stress that the full range of variables 119911 120573 gt 0
is allowedhere because of the hard-core condition (16) it doesnot allow more than 120581♯Λ particles in Λ where ♯Λ stands forthe number of vertices in Λ However while passing to thethermodynamic limit we will need to control 119911 and 120573
Definition 1 We will call GΛand G
Λ|xlowastΓΛ
the Gibbs operatorsin volume Λ for given values of 119911 and 120573 (andmdashin the case ofG
Λ|xlowastΓΛ
mdashwith the boundary condition xlowastΓΛ
)The Gibbs operators in turn give rise to the Gibbs states120593Λ= 120593
120573119911Λand 120593
Λ|xΓΛ
= 120593120573119911Λ|x
ΓΛ
at temperature 120573minus1 andactivity 119911 in volume Λ These are linear positive normalizedfunctionals on the 119862
lowast-algebra BΛof bounded operators in
spaceHΛ
120593Λ(A) = trH
Λ
(RΛA)
120593Λ|xΓΛ
(A) = trHΛ
(RΛ|xΓΛ
A) A isin BΛ
(26)
where
RΛ=
GΛ
Ξ (Λ) with Ξ (Λ) = Ξ
119911120573(Λ) = trH
Λ
GΛ (27)
RΛ|xlowastΓΛ
=
GΛ|xΓΛ
Ξ (Λ | xlowastΓΛ
)
with Ξ (Λ | xlowastΓΛ
) = Ξ119911120573
(Λ | xlowastΓΛ
)
= trHΛ
(119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
])
(28)
The hard-core assumption (16) yields that the quantitiesΞ(Λ) and Ξ
119911120573(Λ | xlowast
ΓΛ) are finite formally these facts will
be verified by virtue of the Feynman-Kac representation
Definition 2 Whenever Λ0sub Λ the 119862lowast-algebraB
Λ0 is iden-
tified with the 119862lowast subalgebra inBΛformed by the operators
of the form A0otimes I
ΛΛ0 Consequently the restriction 120593Λ
0
Λof
state 120593Λto 119862lowast-algebraB
Λ0 is given by
120593Λ0
Λ(A
0) = trH
Λ0
(RΛ0
ΛA0) A
0isin B
Λ0 (29)
where
RΛ0
Λ= trH
ΛΛ0
RΛ (30)
Advances in Mathematical Physics 5
Operators RΛ0
Λ(we again call them RDMs) are positive defi-
nite and have trHΛ0
RΛ0
Λ= 1 They also satisfy the compatibil-
ity property forallΛ0sub Λ
1sub Λ
RΛ0
Λ= trH
Λ1Λ0
RΛ1
Λ (31)
In a similar fashion one defines functionals 120593Λ0
Λ|xlowastΓΛ
and oper-
ators RΛ0
Λ|xlowastΓΛ
with the same properties
15 Limiting Gibbs States The concept of a limiting Gibbsstate is related to notion of a quasilocal 119862lowast-algebra see [14]For the class of systems under consideration the constructionof the quasilocal 119862lowast-algebraB
Γis done along the same lines
as in [1]BΓis the norm completion of the 119862lowast algebra (B0
Γ) =
lim ind119899rarrinfin
BΛ119899
Any family of positive-definite operatorsRΛ0
in spacesHΛ0 of trace one whereΛ0 runs through finite
subsets of Γ with the compatibility property
RΛ1
= trHΛ0Λ1
RΛ0
Λ
1sub Λ
0 (32)
determines a state ofBΓ see [12 13]
Finally we introduce unitary operators UΛ0(g) g isin G in
HΛ0
UΛ120601 (xlowast
Λ0) = 120601 (g
minus1xlowastΛ0) (33)
where
gminus1xlowast
Λ0 = g
minus1xlowast (119895) 119895 isin Λ
0
gminus1xlowast (119895) = g
minus1119909 119909 isin xlowast (119895)
(34)
Theorem3 Assuming the conditions listed above for all 119911 120573 isin
(0 +infin) satisfying (22) and a finite Λ
0sub Γ operators RΛ
0
Λ
form a compact sequence in the trace-norm topology in HΛ0
as Λ Γ Furthermore given any family of (finite or infinite)sets Γ = Γ(Λ) sube Γ and configurations xlowast
ΓΛ= xlowast(119894) 119894 isin Γ Λ
with ♯xlowast(119894) lt 120581 operatorsRΛ0
Λ|xΓΛ
also forma compact sequence
in the trace-norm topology Any limit point RΛ0
for RΛ0
Λ or
RΛ0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operator in H(Λ
0)
of trace one Moreover if Λ1sub Λ
0 and RΛ0
and RΛ1
are thelimits forRΛ
0
ΛandRΛ
1
Λor forRΛ
0
Λ|xlowastΓΛ
andRΛ1
Λ|xlowastΓΛ
along the samesubsequence Λ
119904 Γ then the property (32) holds true
Consequently theGibbs states120593Λand120593
Λ|xlowastΓΛ
form compactsequences as Λ Γ
Remark 4 In fact the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (ΓE)only under an assumption that the degree of the vertices inΓ is uniformy bounded
Definition 5 Any limit point 120593 for states 120593Λand 120593
Λ|xlowastΓΛ
iscalled a limiting Gibbs state (for given 119911 120573 isin (0 +infin))
Theorem 6 Under the condition (22) any limiting pointRΛ0
for RΛ
0
Λ or RΛ
0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operatorof trace one commuting with operators U
Λ0(g) forallg isin G
UΛ0(g)
minus1RΛ0
UΛ0 (g) = RΛ
0
(35)
Accordingly any limiting Gibbs state 120593 of B determined by afamily of limiting operators RΛ
0
obeying (35) satisfies the cor-responding invariance property forall finite Λ0
sub Γ any A isin BΛ0
and g isin G
120593 (A) = 120593 (UΛ0(g)
minus1AUΛ0 (g)) (36)
Remarks (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (ie absence of phase transi-tions)
(2) Properties (35) and (36) are trivially fulfilled for thelimiting points RΛ
0
and 120593 of families RΛ0
Λ and 120593
Λ How-
ever they require a proof for the limit points of the familiesRΛ0
Λ|xlowastΓΛ
and 120593Λ|xlowastΓΛ
The set of limiting Gibbs states (which is nonempty due
to Theorem 3) is denoted by G0 In Section 3 we describe aclassG sup G0 of states of119862lowast-algebraB satisfying the FK-DLRequation similar to that in [1]
2 Feynman-Kac Representations forthe RDM Kernels in a Finite Volume
21 The Representation for the Kernels of the Gibbs OperatorsA starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels K
Λ(xlowast
Λ ylowast
Λ) =
K120573119911Λ
(xlowastΛ ylowast
Λ) and F
Λ(xlowast
Λ ylowast
Λ) = F
120573119911Λ(xlowast
Λ ylowast
Λ) of operators
GΛand R
Λ
Definition 7 Given (119909 119894) (119910 119895) isin 119872 times Γ 119882120573
(119909119894)(119910119895)denotes
the space of path or trajectories 120596 = 120596(119909119894)(119910119895)
in 119872 times Γ oftime-length 120573 with the end-points (119909 119894) and (119910 119895) Formally120596 isin 119882
120573
(119909119894)(119910119895)is defined as follows
120596 120591 isin [0 120573] 997891997888rarr 120596 (120591) = (119906 (120596 120591) 119897 (120596 120591)) isin 119872 times Γ
120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)
120596 has finitely many jumps on [0 120573]
if a jump occurs at time 120591 then d [119897 (120596 120591minus) 119897 (120596 120591)]=1
(37)
The notation 120596(120591) and its alternative (119906(120596 120591) 119897(120596 120591)) forthe position and the index of trajectory 120596 at time 120591 will beemployed as equal in rights We use the term the temporalsection (or simply the section) of path 120596 at time 120591
Definition 8 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ A matching (or pairing) 120574 between xlowast
Λand ylowast
Λis
6 Advances in Mathematical Physics
defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ
119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ
and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast
Λ= ♯ylowast
Λ these properties are equivalent) It is convenient
to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]
Next consider the Cartesian product
119882
120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
120573
(119909119894)120574(119909119894) (38)
and the disjoint union
119882
120573
xlowastΛylowastΛ
= ⋃
120574
119882
120573
xlowastΛylowastΛ120574 (39)
Accordingly an element 120596Λ
isin 119882
120573
xlowastΛylowastΛ120574in (38) represents a
collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573
starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a
path configuration in (or over) Λ
The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation
We will work with standard sigma algebras (generated bycylinder sets) in119882
120573
(119909119894)(119910119895)119882120573
xlowastΛylowastΛ120574 and119882
120573
xlowastΛylowastΛ
Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times
Λ 997891rarr 120595(119909 119894) by
L120595 (119909 119894) = minus
1
2
Δ120595 (119909 119894)
+ sum
119895d(119894119895)=1
120582119894119895int
119872
V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]
(40)
In the probabilistic literature such processes are referred toas Levy processes see for example [14]
Pictorially a trajectory of process 120585 moves along 119872
according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum
119895d(119894119895)=1 120582119894119895 In other words while following a Brownian
motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582
119894119895V(d119910)
After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on
For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573
(119909119894)(119910119895)the nonnormalised measure on 119882
120573
(119909119894)(119910119895)induced
by 120585 That is under measure P120573
(119909119894)(119910119895)the trajectory at time
120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)
= P120573
(119909119894)(119910119895)(119882
120573
(119909119894)(119910119895)) is given by
119901(119909119894)(119910119895)
= 1 (119894 = 119895) 119901
120573
119872(119909 119910) exp[
[
minus120573 sum
119895d(119894119895)=1
120582119894119895]
]
+ sum
119896ge1
sum
1198970=1198941198971119897119896119897119896+1
=119895
prod
0le119904le119896
1 (d (119897119904 119897119904+1
) = 1)
times 120582119897119904119897119904+1
int
120573
0
d120591119904exp[
[
minus (120591119904+1
minus 120591119904) sum
119895d(119897119904119895)=1
120582119897119904119895]
]
times 1 (0 = 1205910lt 120591
1lt sdot sdot sdot lt 120591
119896lt 120591
119896+1= 120573)
(41)
where 119901120573
119872(119909 119910) denotes the transition probability density for
the Brownian motion to pass from 119909 to 119910 on119872 in time 120573
119901
120573
119872(119909 119910) =
1
(2120587120573)
1198892
times sum
119899=(1198991119899119889)isinZ119889
exp(minus
1003816100381610038161003816
119909 minus 119910 + 119899
1003816100381610038161003816
2
2120573
)
(42)
In view of (13) the quantity 119901(119909119894)(119910119895)
and its derivatives areuniformly bounded
119901(119909119894)(119910119895)
10038161003816100381610038161003816
nabla119909119901(119909119894)(119910119895)
10038161003816100381610038161003816
10038161003816100381610038161003816
nabla119910119901(119909119894)(119910119895)
10038161003816100381610038161003816
le 119901119872
119909 119910 isin 119872 119894 119895 isin Γ
(43)
where 119901119872
= 119901119872(120573) isin (0 +infin) is a constant
We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future
Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations overΛ with ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a pairing between xlowast
Λand ylowast
Λ
Then Plowast
xlowastΛylowastΛ120574denotes the product measure on119882
120573
xlowastΛylowastΛ120574
P120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)P120573
(119909119894)120574(119909119894) (44)
Furthermore P120573
xlowastΛylowastΛ
stands for the sum measure on119882
120573
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
= sum
120574
P120573
xlowastΛylowastΛ120574 (45)
According to Definition 10 under the measure P120573
xlowastΛylowastΛ120574
the trajectories 120596119909119894
isin 119882
120573
(119909119894)120574(119909119894)constituting 120596
Λare inde-
pendent components (Here the term independence is usedin the measure-theoretical sense)
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 3
HereΔ(119909)
119895means the Laplacian in variable119909 isin xlowast(119895) Next ♯xlowast
stands for the cardinality of the particle configuration xlowast (ie♯xlowast = 119896 when xlowast isin 119872
(119896)) and the parameter 120581 is introducedin (17) (Symbol ♯ will be used for denoting the cardinalityof a general (finite) set for example ♯Λ means the numberof vertices in Λ) Further xlowast(119895119909)rarr (119895
1015840
119910)
Λdenotes the particle
configuration with the point 119909 isin xlowast(119895) removed and point 119910added to xlowast(1198951015840)
As in [1] we also consider a Hamiltonian HΛ|xlowastΓΛ
in anexternal field generated by a configuration xlowast
ΓΛ= xlowast(1198951015840)
119895
1015840isin Γ Λ isin 119872
lowastΓΛ where Γ sube Γ is a (finite or infinite)collection of vertices More precisely we only consider xlowast
ΓΛ
with ♯xlowast(1198951015840) le 120581 (see (17) below) and set
(HΛ|xlowastΓΛ
120601) (xlowastΛ)
=[
[
minus
1
2
sum
119895isinΛ
sum
119909isinxlowast(119895)Δ
(119909)
119895+ sum
119895isinΛ
sum
119909isinxlowast(119895)119880
(1)(119909)
+
1
2
sum
119895isinΛ
sum
1199091199091015840isinxlowast(119895)
1 (119909 = 119909
1015840)119880
(2)(119909 119909
1015840)
+
1
2
sum
1198951198951015840isinΛ
1 (119895 = 119895
1015840) 119869 (d (119895 119895
1015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)]
]
120601 (xlowastΛ)
+ sum
119895isinΛ1198951015840isinΓΛ
119869 (d (119895 1198951015840))
times sum
119909isinxlowast(119895)1199091015840isinxlowast(1198951015840)
119881 (119909 119909
1015840)120601 (xlowast
Λ)
+ sum
1198951198951015840isinΛ
12058211989511989510158401 (♯xlowast (119895) ge 1 ♯xlowast (1198951015840) lt 120581)
times sum
119909isinxlowast(119895)
int
119872
V (d119910) [120601 (xlowast(119895119909)rarr (1198951015840
119910)
Λ) minus 120601 (xlowast
Λ)]
(12)
The novel elements in (11) and (12) compared with [1]are the presence of on-site potentials 119880(1) and 119880
(2) and thesummand involving transition rates 120582
1198951198951015840 ge 0 for jumps of a
particle from site 119895 to 1198951015840We will suppose that 120582
1198951198951015840 vanishes if the graph distance
d(119895 1198951015840) gt 1 We will also assume uniform boundedness
sup [1205821198951198951015840 (119909119872) 119895 119895
1015840isin Γ 119909 isin 119872] lt infin (13)
in view of (1) it implies that the total exit ratesum
1198951015840d(1198951198951015840)=1 120582119895119895
1015840(119909119872) from site 119895 is uniformly boundedThese conditions are not sharp and can be liberalized
The model under consideration can be considered as ageneralization of the Hubbard model [6] (in its bosonic ver-sion) Its mathematical justification includes the following(a) An opportunity to introduce a Fock space formalismincorporates a number of new features For instance afermonic version of the model (not considered here) emergesnaturally when the bosonic Fock space H(119894) is replaced bya fermonic one Another opening provided by this modelis a possibility to consider random potentials 119880(1) 119880(2) and119881 which would yield a sound generalization of the Mott-Andersonmodel (b) Introducing jumpsmakes a step towardsa treatment of a model of a quantum (Bose-) gas whereparticles ldquoliverdquo in a single Fock space For example a systemof interacting quantum particles is originally confined to aldquoboxrdquo in a Euclidean space with or without ldquointernalrdquo degreesof freedom In the thermodynamical limit the box expandsto the whole Euclidean space In a two-dimensional modelof a quantum gas one expects a phenomenon of invarianceunder space-translations one hopes to be able to address thisissue in future publications (c) A model with jumps can beanalysed by means of the theory of Markov processes whichprovides a developed methodology
Physically speaking the model with jumps covers a situ-ation where ldquolightrdquo quantum particles are subject to a ldquoran-domrdquo force and change their ldquolocationrdquo This class of modelsis interesting from the point of view of transport phenomenathat theymay display (An analogywith the famousAndersonmodel in its multiparticle version inevitably comes tomind cf eg [7]) Methodologically such systems occupyan ldquointermediaterdquo place between models where quantumparticles are ldquofixedrdquo forever to their designated locations (asin [1]) andmodels where quantumparticlesmove in the samespace (a Bose-gas considered in [8 9]) In particular thiswork provides a bridge between [1 8 9] reading this paperahead of [8 9] might help an interested reader to get through[8 9] at a much quicker pace
We would like to note an interesting problem of analysisof the small-mass limit (cf [10]) from the point of Mermin-Wagner phenomena
13 Assumptions on the Potentials The between-sites poten-tial 119881 is assumed to be of class 1198622 Consequently 119881 and itsfirst and second derivatives satisfy uniform bounds Namelyforall119909
1015840 119909
10158401015840isin 119872
minus119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
nablax119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
10038161003816100381610038161003816
nablaxx1015840119881(119909
1015840 119909
10158401015840)
10038161003816100381610038161003816
le 119881 (14)
Here 119909 and 1199091015840 run through the pairs of variables 119909 1199091015840 A simi-lar property is assumed for the on-site potential119880(1) (here weneed only a 1198621 smoothness)
minus119880
(1)(119909)
10038161003816100381610038161003816
nablax119880(1)
(119909)
10038161003816100381610038161003816
le 119880
(1)
119909 isin 119872 (15)
Note that for119881 and119880(1) the bounds are imposed on their neg-ative parts only
As to 119880(2) we suppose that (a)
119880
(2)(119909 119909
1015840) = +infin when 1003816
1003816100381610038161003816
119909 minus 119909
101584010038161003816100381610038161003816
le 120588 (16)
4 Advances in Mathematical Physics
and (b) exist a 1198621-function (119909 119909
1015840) 997891rarr
119880
(2)(119909 119909
1015840) isin R such that
119880
(2)(119909 119909
1015840) =
119880
(2)(119909 119909
1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)
stands for the (flat) Riemannian distance between points119909 119909
1015840isin 119872 As a result of (16) there exists a ldquohard corerdquo of
diameter 120588 and a given atom cannot ldquoholdrdquo more than
120581 = lceil
V (119872)
V (119861 (120588))
rceil (17)
particles where V(119861(120588)) is the volume of a 119889-dimensional ballof diameter 120588 We will also use the bound
minus119880
(2)(119909 119909
1015840)
10038161003816100381610038161003816
nablax119880(2)
(119909 119909
1015840)
10038161003816100381610038161003816
le 119880
(2)
119909 119909
1015840isin 119872 (18)
Formally (16) means that the operators in (11) and (12) areconsidered for functions120601(xlowast
Λ) vanishingwhen in the particle
configuration xlowastΛ= xlowast(119895) 119895 isin Λ the cardinality ♯xlowast(119895) gt 120581
for some 119895 isin ΛThe function 119869 119903 isin (0infin) 997891rarr 119869(119903) ge 0 is assumed
monotonically nonincreasing with 119903 and obeying the relation119869(119897) rarr 0 as 119897 rarr infin where
119869 (119897) = sup[
[
sum
11989510158401015840isinΓ
119869 (d (1198951015840 119895
10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895
1015840isin Γ
]
]
lt infin
(19)
Additionally let 119869(119903) be such that
119869
lowast= sup[
[
sum
1198951015840isinΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
119895 isin Γ]
]
lt infin (20)
Next we assume that the functions 119880(1) 119880(2) and 119881 are g-invariant forall119909 1199091015840 isin 119872 and g isin G
119880
(1)(119909) = 119880
(1)(g119909)
119880
(2)(119909 119909
1015840) = 119880
(2)(g119909 g119909
1015840)
119881 (119909 119909
1015840) = 119881 (g119909 g119909
1015840)
(21)
In the following we will need to bound the fugacity (oractivity cf (25)) 119911 in terms of the other parameters of themodel
119911119890
Θlt 1 where Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881) (22)
14 The Gibbs State in a Finite Volume Define the particlenumber operator N
Λ with the action
NΛ120601 (xlowast
Λ) = ♯xlowast
Λ120601 (xlowast
Λ) xlowast
Λisin 119872
lowastΛ (23)
Here for a given xlowastΛ= xlowast(119895) 119895 isin Λ ♯xlowast
Λstands for the total
number of particles in configuration xlowastΛ
♯xlowastΛ= sum
119895isinΛ
♯xlowast (119895) (24)
The standard canonical variable associated withNΛis activity
119911 isin (0infin)The Hamiltonians (11) and (12) are self-adjoint (on the
natural domains) in H(Λ) Moreover they are positivedefinite and have a discrete spectrum cf [14] Furthermoreforall119911 120573 gt 0 H
Λand H
Λ|xlowastΓΛ
give rise to positive-definite trace-class operators G
Λ= G
119911120573Λand G
Λ|xlowastΓΛ
= G119911120573Λ|xlowast
ΓΛ
GΛ= 119911
NΛ exp [minus120573H
Λ]
GΛ|xlowastΓΛ
= 119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
]
(25)
We would like to stress that the full range of variables 119911 120573 gt 0
is allowedhere because of the hard-core condition (16) it doesnot allow more than 120581♯Λ particles in Λ where ♯Λ stands forthe number of vertices in Λ However while passing to thethermodynamic limit we will need to control 119911 and 120573
Definition 1 We will call GΛand G
Λ|xlowastΓΛ
the Gibbs operatorsin volume Λ for given values of 119911 and 120573 (andmdashin the case ofG
Λ|xlowastΓΛ
mdashwith the boundary condition xlowastΓΛ
)The Gibbs operators in turn give rise to the Gibbs states120593Λ= 120593
120573119911Λand 120593
Λ|xΓΛ
= 120593120573119911Λ|x
ΓΛ
at temperature 120573minus1 andactivity 119911 in volume Λ These are linear positive normalizedfunctionals on the 119862
lowast-algebra BΛof bounded operators in
spaceHΛ
120593Λ(A) = trH
Λ
(RΛA)
120593Λ|xΓΛ
(A) = trHΛ
(RΛ|xΓΛ
A) A isin BΛ
(26)
where
RΛ=
GΛ
Ξ (Λ) with Ξ (Λ) = Ξ
119911120573(Λ) = trH
Λ
GΛ (27)
RΛ|xlowastΓΛ
=
GΛ|xΓΛ
Ξ (Λ | xlowastΓΛ
)
with Ξ (Λ | xlowastΓΛ
) = Ξ119911120573
(Λ | xlowastΓΛ
)
= trHΛ
(119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
])
(28)
The hard-core assumption (16) yields that the quantitiesΞ(Λ) and Ξ
119911120573(Λ | xlowast
ΓΛ) are finite formally these facts will
be verified by virtue of the Feynman-Kac representation
Definition 2 Whenever Λ0sub Λ the 119862lowast-algebraB
Λ0 is iden-
tified with the 119862lowast subalgebra inBΛformed by the operators
of the form A0otimes I
ΛΛ0 Consequently the restriction 120593Λ
0
Λof
state 120593Λto 119862lowast-algebraB
Λ0 is given by
120593Λ0
Λ(A
0) = trH
Λ0
(RΛ0
ΛA0) A
0isin B
Λ0 (29)
where
RΛ0
Λ= trH
ΛΛ0
RΛ (30)
Advances in Mathematical Physics 5
Operators RΛ0
Λ(we again call them RDMs) are positive defi-
nite and have trHΛ0
RΛ0
Λ= 1 They also satisfy the compatibil-
ity property forallΛ0sub Λ
1sub Λ
RΛ0
Λ= trH
Λ1Λ0
RΛ1
Λ (31)
In a similar fashion one defines functionals 120593Λ0
Λ|xlowastΓΛ
and oper-
ators RΛ0
Λ|xlowastΓΛ
with the same properties
15 Limiting Gibbs States The concept of a limiting Gibbsstate is related to notion of a quasilocal 119862lowast-algebra see [14]For the class of systems under consideration the constructionof the quasilocal 119862lowast-algebraB
Γis done along the same lines
as in [1]BΓis the norm completion of the 119862lowast algebra (B0
Γ) =
lim ind119899rarrinfin
BΛ119899
Any family of positive-definite operatorsRΛ0
in spacesHΛ0 of trace one whereΛ0 runs through finite
subsets of Γ with the compatibility property
RΛ1
= trHΛ0Λ1
RΛ0
Λ
1sub Λ
0 (32)
determines a state ofBΓ see [12 13]
Finally we introduce unitary operators UΛ0(g) g isin G in
HΛ0
UΛ120601 (xlowast
Λ0) = 120601 (g
minus1xlowastΛ0) (33)
where
gminus1xlowast
Λ0 = g
minus1xlowast (119895) 119895 isin Λ
0
gminus1xlowast (119895) = g
minus1119909 119909 isin xlowast (119895)
(34)
Theorem3 Assuming the conditions listed above for all 119911 120573 isin
(0 +infin) satisfying (22) and a finite Λ
0sub Γ operators RΛ
0
Λ
form a compact sequence in the trace-norm topology in HΛ0
as Λ Γ Furthermore given any family of (finite or infinite)sets Γ = Γ(Λ) sube Γ and configurations xlowast
ΓΛ= xlowast(119894) 119894 isin Γ Λ
with ♯xlowast(119894) lt 120581 operatorsRΛ0
Λ|xΓΛ
also forma compact sequence
in the trace-norm topology Any limit point RΛ0
for RΛ0
Λ or
RΛ0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operator in H(Λ
0)
of trace one Moreover if Λ1sub Λ
0 and RΛ0
and RΛ1
are thelimits forRΛ
0
ΛandRΛ
1
Λor forRΛ
0
Λ|xlowastΓΛ
andRΛ1
Λ|xlowastΓΛ
along the samesubsequence Λ
119904 Γ then the property (32) holds true
Consequently theGibbs states120593Λand120593
Λ|xlowastΓΛ
form compactsequences as Λ Γ
Remark 4 In fact the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (ΓE)only under an assumption that the degree of the vertices inΓ is uniformy bounded
Definition 5 Any limit point 120593 for states 120593Λand 120593
Λ|xlowastΓΛ
iscalled a limiting Gibbs state (for given 119911 120573 isin (0 +infin))
Theorem 6 Under the condition (22) any limiting pointRΛ0
for RΛ
0
Λ or RΛ
0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operatorof trace one commuting with operators U
Λ0(g) forallg isin G
UΛ0(g)
minus1RΛ0
UΛ0 (g) = RΛ
0
(35)
Accordingly any limiting Gibbs state 120593 of B determined by afamily of limiting operators RΛ
0
obeying (35) satisfies the cor-responding invariance property forall finite Λ0
sub Γ any A isin BΛ0
and g isin G
120593 (A) = 120593 (UΛ0(g)
minus1AUΛ0 (g)) (36)
Remarks (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (ie absence of phase transi-tions)
(2) Properties (35) and (36) are trivially fulfilled for thelimiting points RΛ
0
and 120593 of families RΛ0
Λ and 120593
Λ How-
ever they require a proof for the limit points of the familiesRΛ0
Λ|xlowastΓΛ
and 120593Λ|xlowastΓΛ
The set of limiting Gibbs states (which is nonempty due
to Theorem 3) is denoted by G0 In Section 3 we describe aclassG sup G0 of states of119862lowast-algebraB satisfying the FK-DLRequation similar to that in [1]
2 Feynman-Kac Representations forthe RDM Kernels in a Finite Volume
21 The Representation for the Kernels of the Gibbs OperatorsA starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels K
Λ(xlowast
Λ ylowast
Λ) =
K120573119911Λ
(xlowastΛ ylowast
Λ) and F
Λ(xlowast
Λ ylowast
Λ) = F
120573119911Λ(xlowast
Λ ylowast
Λ) of operators
GΛand R
Λ
Definition 7 Given (119909 119894) (119910 119895) isin 119872 times Γ 119882120573
(119909119894)(119910119895)denotes
the space of path or trajectories 120596 = 120596(119909119894)(119910119895)
in 119872 times Γ oftime-length 120573 with the end-points (119909 119894) and (119910 119895) Formally120596 isin 119882
120573
(119909119894)(119910119895)is defined as follows
120596 120591 isin [0 120573] 997891997888rarr 120596 (120591) = (119906 (120596 120591) 119897 (120596 120591)) isin 119872 times Γ
120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)
120596 has finitely many jumps on [0 120573]
if a jump occurs at time 120591 then d [119897 (120596 120591minus) 119897 (120596 120591)]=1
(37)
The notation 120596(120591) and its alternative (119906(120596 120591) 119897(120596 120591)) forthe position and the index of trajectory 120596 at time 120591 will beemployed as equal in rights We use the term the temporalsection (or simply the section) of path 120596 at time 120591
Definition 8 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ A matching (or pairing) 120574 between xlowast
Λand ylowast
Λis
6 Advances in Mathematical Physics
defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ
119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ
and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast
Λ= ♯ylowast
Λ these properties are equivalent) It is convenient
to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]
Next consider the Cartesian product
119882
120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
120573
(119909119894)120574(119909119894) (38)
and the disjoint union
119882
120573
xlowastΛylowastΛ
= ⋃
120574
119882
120573
xlowastΛylowastΛ120574 (39)
Accordingly an element 120596Λ
isin 119882
120573
xlowastΛylowastΛ120574in (38) represents a
collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573
starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a
path configuration in (or over) Λ
The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation
We will work with standard sigma algebras (generated bycylinder sets) in119882
120573
(119909119894)(119910119895)119882120573
xlowastΛylowastΛ120574 and119882
120573
xlowastΛylowastΛ
Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times
Λ 997891rarr 120595(119909 119894) by
L120595 (119909 119894) = minus
1
2
Δ120595 (119909 119894)
+ sum
119895d(119894119895)=1
120582119894119895int
119872
V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]
(40)
In the probabilistic literature such processes are referred toas Levy processes see for example [14]
Pictorially a trajectory of process 120585 moves along 119872
according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum
119895d(119894119895)=1 120582119894119895 In other words while following a Brownian
motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582
119894119895V(d119910)
After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on
For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573
(119909119894)(119910119895)the nonnormalised measure on 119882
120573
(119909119894)(119910119895)induced
by 120585 That is under measure P120573
(119909119894)(119910119895)the trajectory at time
120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)
= P120573
(119909119894)(119910119895)(119882
120573
(119909119894)(119910119895)) is given by
119901(119909119894)(119910119895)
= 1 (119894 = 119895) 119901
120573
119872(119909 119910) exp[
[
minus120573 sum
119895d(119894119895)=1
120582119894119895]
]
+ sum
119896ge1
sum
1198970=1198941198971119897119896119897119896+1
=119895
prod
0le119904le119896
1 (d (119897119904 119897119904+1
) = 1)
times 120582119897119904119897119904+1
int
120573
0
d120591119904exp[
[
minus (120591119904+1
minus 120591119904) sum
119895d(119897119904119895)=1
120582119897119904119895]
]
times 1 (0 = 1205910lt 120591
1lt sdot sdot sdot lt 120591
119896lt 120591
119896+1= 120573)
(41)
where 119901120573
119872(119909 119910) denotes the transition probability density for
the Brownian motion to pass from 119909 to 119910 on119872 in time 120573
119901
120573
119872(119909 119910) =
1
(2120587120573)
1198892
times sum
119899=(1198991119899119889)isinZ119889
exp(minus
1003816100381610038161003816
119909 minus 119910 + 119899
1003816100381610038161003816
2
2120573
)
(42)
In view of (13) the quantity 119901(119909119894)(119910119895)
and its derivatives areuniformly bounded
119901(119909119894)(119910119895)
10038161003816100381610038161003816
nabla119909119901(119909119894)(119910119895)
10038161003816100381610038161003816
10038161003816100381610038161003816
nabla119910119901(119909119894)(119910119895)
10038161003816100381610038161003816
le 119901119872
119909 119910 isin 119872 119894 119895 isin Γ
(43)
where 119901119872
= 119901119872(120573) isin (0 +infin) is a constant
We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future
Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations overΛ with ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a pairing between xlowast
Λand ylowast
Λ
Then Plowast
xlowastΛylowastΛ120574denotes the product measure on119882
120573
xlowastΛylowastΛ120574
P120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)P120573
(119909119894)120574(119909119894) (44)
Furthermore P120573
xlowastΛylowastΛ
stands for the sum measure on119882
120573
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
= sum
120574
P120573
xlowastΛylowastΛ120574 (45)
According to Definition 10 under the measure P120573
xlowastΛylowastΛ120574
the trajectories 120596119909119894
isin 119882
120573
(119909119894)120574(119909119894)constituting 120596
Λare inde-
pendent components (Here the term independence is usedin the measure-theoretical sense)
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Mathematical Physics
and (b) exist a 1198621-function (119909 119909
1015840) 997891rarr
119880
(2)(119909 119909
1015840) isin R such that
119880
(2)(119909 119909
1015840) =
119880
(2)(119909 119909
1015840) whenever 120588(119909 1199091015840) gt 120588 Here 120588(119909 1199091015840)
stands for the (flat) Riemannian distance between points119909 119909
1015840isin 119872 As a result of (16) there exists a ldquohard corerdquo of
diameter 120588 and a given atom cannot ldquoholdrdquo more than
120581 = lceil
V (119872)
V (119861 (120588))
rceil (17)
particles where V(119861(120588)) is the volume of a 119889-dimensional ballof diameter 120588 We will also use the bound
minus119880
(2)(119909 119909
1015840)
10038161003816100381610038161003816
nablax119880(2)
(119909 119909
1015840)
10038161003816100381610038161003816
le 119880
(2)
119909 119909
1015840isin 119872 (18)
Formally (16) means that the operators in (11) and (12) areconsidered for functions120601(xlowast
Λ) vanishingwhen in the particle
configuration xlowastΛ= xlowast(119895) 119895 isin Λ the cardinality ♯xlowast(119895) gt 120581
for some 119895 isin ΛThe function 119869 119903 isin (0infin) 997891rarr 119869(119903) ge 0 is assumed
monotonically nonincreasing with 119903 and obeying the relation119869(119897) rarr 0 as 119897 rarr infin where
119869 (119897) = sup[
[
sum
11989510158401015840isinΓ
119869 (d (1198951015840 119895
10158401015840)) 1 (d (1198951015840 11989510158401015840) ge 119897) 119895
1015840isin Γ
]
]
lt infin
(19)
Additionally let 119869(119903) be such that
119869
lowast= sup[
[
sum
1198951015840isinΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
119895 isin Γ]
]
lt infin (20)
Next we assume that the functions 119880(1) 119880(2) and 119881 are g-invariant forall119909 1199091015840 isin 119872 and g isin G
119880
(1)(119909) = 119880
(1)(g119909)
119880
(2)(119909 119909
1015840) = 119880
(2)(g119909 g119909
1015840)
119881 (119909 119909
1015840) = 119881 (g119909 g119909
1015840)
(21)
In the following we will need to bound the fugacity (oractivity cf (25)) 119911 in terms of the other parameters of themodel
119911119890
Θlt 1 where Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881) (22)
14 The Gibbs State in a Finite Volume Define the particlenumber operator N
Λ with the action
NΛ120601 (xlowast
Λ) = ♯xlowast
Λ120601 (xlowast
Λ) xlowast
Λisin 119872
lowastΛ (23)
Here for a given xlowastΛ= xlowast(119895) 119895 isin Λ ♯xlowast
Λstands for the total
number of particles in configuration xlowastΛ
♯xlowastΛ= sum
119895isinΛ
♯xlowast (119895) (24)
The standard canonical variable associated withNΛis activity
119911 isin (0infin)The Hamiltonians (11) and (12) are self-adjoint (on the
natural domains) in H(Λ) Moreover they are positivedefinite and have a discrete spectrum cf [14] Furthermoreforall119911 120573 gt 0 H
Λand H
Λ|xlowastΓΛ
give rise to positive-definite trace-class operators G
Λ= G
119911120573Λand G
Λ|xlowastΓΛ
= G119911120573Λ|xlowast
ΓΛ
GΛ= 119911
NΛ exp [minus120573H
Λ]
GΛ|xlowastΓΛ
= 119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
]
(25)
We would like to stress that the full range of variables 119911 120573 gt 0
is allowedhere because of the hard-core condition (16) it doesnot allow more than 120581♯Λ particles in Λ where ♯Λ stands forthe number of vertices in Λ However while passing to thethermodynamic limit we will need to control 119911 and 120573
Definition 1 We will call GΛand G
Λ|xlowastΓΛ
the Gibbs operatorsin volume Λ for given values of 119911 and 120573 (andmdashin the case ofG
Λ|xlowastΓΛ
mdashwith the boundary condition xlowastΓΛ
)The Gibbs operators in turn give rise to the Gibbs states120593Λ= 120593
120573119911Λand 120593
Λ|xΓΛ
= 120593120573119911Λ|x
ΓΛ
at temperature 120573minus1 andactivity 119911 in volume Λ These are linear positive normalizedfunctionals on the 119862
lowast-algebra BΛof bounded operators in
spaceHΛ
120593Λ(A) = trH
Λ
(RΛA)
120593Λ|xΓΛ
(A) = trHΛ
(RΛ|xΓΛ
A) A isin BΛ
(26)
where
RΛ=
GΛ
Ξ (Λ) with Ξ (Λ) = Ξ
119911120573(Λ) = trH
Λ
GΛ (27)
RΛ|xlowastΓΛ
=
GΛ|xΓΛ
Ξ (Λ | xlowastΓΛ
)
with Ξ (Λ | xlowastΓΛ
) = Ξ119911120573
(Λ | xlowastΓΛ
)
= trHΛ
(119911
NΛ exp [minus120573H
Λ|xlowastΓΛ
])
(28)
The hard-core assumption (16) yields that the quantitiesΞ(Λ) and Ξ
119911120573(Λ | xlowast
ΓΛ) are finite formally these facts will
be verified by virtue of the Feynman-Kac representation
Definition 2 Whenever Λ0sub Λ the 119862lowast-algebraB
Λ0 is iden-
tified with the 119862lowast subalgebra inBΛformed by the operators
of the form A0otimes I
ΛΛ0 Consequently the restriction 120593Λ
0
Λof
state 120593Λto 119862lowast-algebraB
Λ0 is given by
120593Λ0
Λ(A
0) = trH
Λ0
(RΛ0
ΛA0) A
0isin B
Λ0 (29)
where
RΛ0
Λ= trH
ΛΛ0
RΛ (30)
Advances in Mathematical Physics 5
Operators RΛ0
Λ(we again call them RDMs) are positive defi-
nite and have trHΛ0
RΛ0
Λ= 1 They also satisfy the compatibil-
ity property forallΛ0sub Λ
1sub Λ
RΛ0
Λ= trH
Λ1Λ0
RΛ1
Λ (31)
In a similar fashion one defines functionals 120593Λ0
Λ|xlowastΓΛ
and oper-
ators RΛ0
Λ|xlowastΓΛ
with the same properties
15 Limiting Gibbs States The concept of a limiting Gibbsstate is related to notion of a quasilocal 119862lowast-algebra see [14]For the class of systems under consideration the constructionof the quasilocal 119862lowast-algebraB
Γis done along the same lines
as in [1]BΓis the norm completion of the 119862lowast algebra (B0
Γ) =
lim ind119899rarrinfin
BΛ119899
Any family of positive-definite operatorsRΛ0
in spacesHΛ0 of trace one whereΛ0 runs through finite
subsets of Γ with the compatibility property
RΛ1
= trHΛ0Λ1
RΛ0
Λ
1sub Λ
0 (32)
determines a state ofBΓ see [12 13]
Finally we introduce unitary operators UΛ0(g) g isin G in
HΛ0
UΛ120601 (xlowast
Λ0) = 120601 (g
minus1xlowastΛ0) (33)
where
gminus1xlowast
Λ0 = g
minus1xlowast (119895) 119895 isin Λ
0
gminus1xlowast (119895) = g
minus1119909 119909 isin xlowast (119895)
(34)
Theorem3 Assuming the conditions listed above for all 119911 120573 isin
(0 +infin) satisfying (22) and a finite Λ
0sub Γ operators RΛ
0
Λ
form a compact sequence in the trace-norm topology in HΛ0
as Λ Γ Furthermore given any family of (finite or infinite)sets Γ = Γ(Λ) sube Γ and configurations xlowast
ΓΛ= xlowast(119894) 119894 isin Γ Λ
with ♯xlowast(119894) lt 120581 operatorsRΛ0
Λ|xΓΛ
also forma compact sequence
in the trace-norm topology Any limit point RΛ0
for RΛ0
Λ or
RΛ0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operator in H(Λ
0)
of trace one Moreover if Λ1sub Λ
0 and RΛ0
and RΛ1
are thelimits forRΛ
0
ΛandRΛ
1
Λor forRΛ
0
Λ|xlowastΓΛ
andRΛ1
Λ|xlowastΓΛ
along the samesubsequence Λ
119904 Γ then the property (32) holds true
Consequently theGibbs states120593Λand120593
Λ|xlowastΓΛ
form compactsequences as Λ Γ
Remark 4 In fact the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (ΓE)only under an assumption that the degree of the vertices inΓ is uniformy bounded
Definition 5 Any limit point 120593 for states 120593Λand 120593
Λ|xlowastΓΛ
iscalled a limiting Gibbs state (for given 119911 120573 isin (0 +infin))
Theorem 6 Under the condition (22) any limiting pointRΛ0
for RΛ
0
Λ or RΛ
0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operatorof trace one commuting with operators U
Λ0(g) forallg isin G
UΛ0(g)
minus1RΛ0
UΛ0 (g) = RΛ
0
(35)
Accordingly any limiting Gibbs state 120593 of B determined by afamily of limiting operators RΛ
0
obeying (35) satisfies the cor-responding invariance property forall finite Λ0
sub Γ any A isin BΛ0
and g isin G
120593 (A) = 120593 (UΛ0(g)
minus1AUΛ0 (g)) (36)
Remarks (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (ie absence of phase transi-tions)
(2) Properties (35) and (36) are trivially fulfilled for thelimiting points RΛ
0
and 120593 of families RΛ0
Λ and 120593
Λ How-
ever they require a proof for the limit points of the familiesRΛ0
Λ|xlowastΓΛ
and 120593Λ|xlowastΓΛ
The set of limiting Gibbs states (which is nonempty due
to Theorem 3) is denoted by G0 In Section 3 we describe aclassG sup G0 of states of119862lowast-algebraB satisfying the FK-DLRequation similar to that in [1]
2 Feynman-Kac Representations forthe RDM Kernels in a Finite Volume
21 The Representation for the Kernels of the Gibbs OperatorsA starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels K
Λ(xlowast
Λ ylowast
Λ) =
K120573119911Λ
(xlowastΛ ylowast
Λ) and F
Λ(xlowast
Λ ylowast
Λ) = F
120573119911Λ(xlowast
Λ ylowast
Λ) of operators
GΛand R
Λ
Definition 7 Given (119909 119894) (119910 119895) isin 119872 times Γ 119882120573
(119909119894)(119910119895)denotes
the space of path or trajectories 120596 = 120596(119909119894)(119910119895)
in 119872 times Γ oftime-length 120573 with the end-points (119909 119894) and (119910 119895) Formally120596 isin 119882
120573
(119909119894)(119910119895)is defined as follows
120596 120591 isin [0 120573] 997891997888rarr 120596 (120591) = (119906 (120596 120591) 119897 (120596 120591)) isin 119872 times Γ
120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)
120596 has finitely many jumps on [0 120573]
if a jump occurs at time 120591 then d [119897 (120596 120591minus) 119897 (120596 120591)]=1
(37)
The notation 120596(120591) and its alternative (119906(120596 120591) 119897(120596 120591)) forthe position and the index of trajectory 120596 at time 120591 will beemployed as equal in rights We use the term the temporalsection (or simply the section) of path 120596 at time 120591
Definition 8 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ A matching (or pairing) 120574 between xlowast
Λand ylowast
Λis
6 Advances in Mathematical Physics
defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ
119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ
and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast
Λ= ♯ylowast
Λ these properties are equivalent) It is convenient
to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]
Next consider the Cartesian product
119882
120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
120573
(119909119894)120574(119909119894) (38)
and the disjoint union
119882
120573
xlowastΛylowastΛ
= ⋃
120574
119882
120573
xlowastΛylowastΛ120574 (39)
Accordingly an element 120596Λ
isin 119882
120573
xlowastΛylowastΛ120574in (38) represents a
collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573
starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a
path configuration in (or over) Λ
The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation
We will work with standard sigma algebras (generated bycylinder sets) in119882
120573
(119909119894)(119910119895)119882120573
xlowastΛylowastΛ120574 and119882
120573
xlowastΛylowastΛ
Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times
Λ 997891rarr 120595(119909 119894) by
L120595 (119909 119894) = minus
1
2
Δ120595 (119909 119894)
+ sum
119895d(119894119895)=1
120582119894119895int
119872
V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]
(40)
In the probabilistic literature such processes are referred toas Levy processes see for example [14]
Pictorially a trajectory of process 120585 moves along 119872
according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum
119895d(119894119895)=1 120582119894119895 In other words while following a Brownian
motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582
119894119895V(d119910)
After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on
For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573
(119909119894)(119910119895)the nonnormalised measure on 119882
120573
(119909119894)(119910119895)induced
by 120585 That is under measure P120573
(119909119894)(119910119895)the trajectory at time
120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)
= P120573
(119909119894)(119910119895)(119882
120573
(119909119894)(119910119895)) is given by
119901(119909119894)(119910119895)
= 1 (119894 = 119895) 119901
120573
119872(119909 119910) exp[
[
minus120573 sum
119895d(119894119895)=1
120582119894119895]
]
+ sum
119896ge1
sum
1198970=1198941198971119897119896119897119896+1
=119895
prod
0le119904le119896
1 (d (119897119904 119897119904+1
) = 1)
times 120582119897119904119897119904+1
int
120573
0
d120591119904exp[
[
minus (120591119904+1
minus 120591119904) sum
119895d(119897119904119895)=1
120582119897119904119895]
]
times 1 (0 = 1205910lt 120591
1lt sdot sdot sdot lt 120591
119896lt 120591
119896+1= 120573)
(41)
where 119901120573
119872(119909 119910) denotes the transition probability density for
the Brownian motion to pass from 119909 to 119910 on119872 in time 120573
119901
120573
119872(119909 119910) =
1
(2120587120573)
1198892
times sum
119899=(1198991119899119889)isinZ119889
exp(minus
1003816100381610038161003816
119909 minus 119910 + 119899
1003816100381610038161003816
2
2120573
)
(42)
In view of (13) the quantity 119901(119909119894)(119910119895)
and its derivatives areuniformly bounded
119901(119909119894)(119910119895)
10038161003816100381610038161003816
nabla119909119901(119909119894)(119910119895)
10038161003816100381610038161003816
10038161003816100381610038161003816
nabla119910119901(119909119894)(119910119895)
10038161003816100381610038161003816
le 119901119872
119909 119910 isin 119872 119894 119895 isin Γ
(43)
where 119901119872
= 119901119872(120573) isin (0 +infin) is a constant
We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future
Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations overΛ with ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a pairing between xlowast
Λand ylowast
Λ
Then Plowast
xlowastΛylowastΛ120574denotes the product measure on119882
120573
xlowastΛylowastΛ120574
P120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)P120573
(119909119894)120574(119909119894) (44)
Furthermore P120573
xlowastΛylowastΛ
stands for the sum measure on119882
120573
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
= sum
120574
P120573
xlowastΛylowastΛ120574 (45)
According to Definition 10 under the measure P120573
xlowastΛylowastΛ120574
the trajectories 120596119909119894
isin 119882
120573
(119909119894)120574(119909119894)constituting 120596
Λare inde-
pendent components (Here the term independence is usedin the measure-theoretical sense)
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 5
Operators RΛ0
Λ(we again call them RDMs) are positive defi-
nite and have trHΛ0
RΛ0
Λ= 1 They also satisfy the compatibil-
ity property forallΛ0sub Λ
1sub Λ
RΛ0
Λ= trH
Λ1Λ0
RΛ1
Λ (31)
In a similar fashion one defines functionals 120593Λ0
Λ|xlowastΓΛ
and oper-
ators RΛ0
Λ|xlowastΓΛ
with the same properties
15 Limiting Gibbs States The concept of a limiting Gibbsstate is related to notion of a quasilocal 119862lowast-algebra see [14]For the class of systems under consideration the constructionof the quasilocal 119862lowast-algebraB
Γis done along the same lines
as in [1]BΓis the norm completion of the 119862lowast algebra (B0
Γ) =
lim ind119899rarrinfin
BΛ119899
Any family of positive-definite operatorsRΛ0
in spacesHΛ0 of trace one whereΛ0 runs through finite
subsets of Γ with the compatibility property
RΛ1
= trHΛ0Λ1
RΛ0
Λ
1sub Λ
0 (32)
determines a state ofBΓ see [12 13]
Finally we introduce unitary operators UΛ0(g) g isin G in
HΛ0
UΛ120601 (xlowast
Λ0) = 120601 (g
minus1xlowastΛ0) (33)
where
gminus1xlowast
Λ0 = g
minus1xlowast (119895) 119895 isin Λ
0
gminus1xlowast (119895) = g
minus1119909 119909 isin xlowast (119895)
(34)
Theorem3 Assuming the conditions listed above for all 119911 120573 isin
(0 +infin) satisfying (22) and a finite Λ
0sub Γ operators RΛ
0
Λ
form a compact sequence in the trace-norm topology in HΛ0
as Λ Γ Furthermore given any family of (finite or infinite)sets Γ = Γ(Λ) sube Γ and configurations xlowast
ΓΛ= xlowast(119894) 119894 isin Γ Λ
with ♯xlowast(119894) lt 120581 operatorsRΛ0
Λ|xΓΛ
also forma compact sequence
in the trace-norm topology Any limit point RΛ0
for RΛ0
Λ or
RΛ0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operator in H(Λ
0)
of trace one Moreover if Λ1sub Λ
0 and RΛ0
and RΛ1
are thelimits forRΛ
0
ΛandRΛ
1
Λor forRΛ
0
Λ|xlowastΓΛ
andRΛ1
Λ|xlowastΓΛ
along the samesubsequence Λ
119904 Γ then the property (32) holds true
Consequently theGibbs states120593Λand120593
Λ|xlowastΓΛ
form compactsequences as Λ Γ
Remark 4 In fact the assertion of Theorem 3 holds withoutassuming the bidimensionality condition on graph (ΓE)only under an assumption that the degree of the vertices inΓ is uniformy bounded
Definition 5 Any limit point 120593 for states 120593Λand 120593
Λ|xlowastΓΛ
iscalled a limiting Gibbs state (for given 119911 120573 isin (0 +infin))
Theorem 6 Under the condition (22) any limiting pointRΛ0
for RΛ
0
Λ or RΛ
0
Λ|xlowastΓΛ
as Λ Γ is a positive-definite operatorof trace one commuting with operators U
Λ0(g) forallg isin G
UΛ0(g)
minus1RΛ0
UΛ0 (g) = RΛ
0
(35)
Accordingly any limiting Gibbs state 120593 of B determined by afamily of limiting operators RΛ
0
obeying (35) satisfies the cor-responding invariance property forall finite Λ0
sub Γ any A isin BΛ0
and g isin G
120593 (A) = 120593 (UΛ0(g)
minus1AUΛ0 (g)) (36)
Remarks (1) Condition (22) does not imply the uniqueness ofan infinite-volume Gibbs state (ie absence of phase transi-tions)
(2) Properties (35) and (36) are trivially fulfilled for thelimiting points RΛ
0
and 120593 of families RΛ0
Λ and 120593
Λ How-
ever they require a proof for the limit points of the familiesRΛ0
Λ|xlowastΓΛ
and 120593Λ|xlowastΓΛ
The set of limiting Gibbs states (which is nonempty due
to Theorem 3) is denoted by G0 In Section 3 we describe aclassG sup G0 of states of119862lowast-algebraB satisfying the FK-DLRequation similar to that in [1]
2 Feynman-Kac Representations forthe RDM Kernels in a Finite Volume
21 The Representation for the Kernels of the Gibbs OperatorsA starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels K
Λ(xlowast
Λ ylowast
Λ) =
K120573119911Λ
(xlowastΛ ylowast
Λ) and F
Λ(xlowast
Λ ylowast
Λ) = F
120573119911Λ(xlowast
Λ ylowast
Λ) of operators
GΛand R
Λ
Definition 7 Given (119909 119894) (119910 119895) isin 119872 times Γ 119882120573
(119909119894)(119910119895)denotes
the space of path or trajectories 120596 = 120596(119909119894)(119910119895)
in 119872 times Γ oftime-length 120573 with the end-points (119909 119894) and (119910 119895) Formally120596 isin 119882
120573
(119909119894)(119910119895)is defined as follows
120596 120591 isin [0 120573] 997891997888rarr 120596 (120591) = (119906 (120596 120591) 119897 (120596 120591)) isin 119872 times Γ
120596 is cadlag 120596 (0) = (119909 119894) 120596 (120573minus) = (119910 119895)
120596 has finitely many jumps on [0 120573]
if a jump occurs at time 120591 then d [119897 (120596 120591minus) 119897 (120596 120591)]=1
(37)
The notation 120596(120591) and its alternative (119906(120596 120591) 119897(120596 120591)) forthe position and the index of trajectory 120596 at time 120591 will beemployed as equal in rights We use the term the temporalsection (or simply the section) of path 120596 at time 120591
Definition 8 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ A matching (or pairing) 120574 between xlowast
Λand ylowast
Λis
6 Advances in Mathematical Physics
defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ
119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ
and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast
Λ= ♯ylowast
Λ these properties are equivalent) It is convenient
to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]
Next consider the Cartesian product
119882
120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
120573
(119909119894)120574(119909119894) (38)
and the disjoint union
119882
120573
xlowastΛylowastΛ
= ⋃
120574
119882
120573
xlowastΛylowastΛ120574 (39)
Accordingly an element 120596Λ
isin 119882
120573
xlowastΛylowastΛ120574in (38) represents a
collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573
starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a
path configuration in (or over) Λ
The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation
We will work with standard sigma algebras (generated bycylinder sets) in119882
120573
(119909119894)(119910119895)119882120573
xlowastΛylowastΛ120574 and119882
120573
xlowastΛylowastΛ
Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times
Λ 997891rarr 120595(119909 119894) by
L120595 (119909 119894) = minus
1
2
Δ120595 (119909 119894)
+ sum
119895d(119894119895)=1
120582119894119895int
119872
V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]
(40)
In the probabilistic literature such processes are referred toas Levy processes see for example [14]
Pictorially a trajectory of process 120585 moves along 119872
according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum
119895d(119894119895)=1 120582119894119895 In other words while following a Brownian
motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582
119894119895V(d119910)
After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on
For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573
(119909119894)(119910119895)the nonnormalised measure on 119882
120573
(119909119894)(119910119895)induced
by 120585 That is under measure P120573
(119909119894)(119910119895)the trajectory at time
120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)
= P120573
(119909119894)(119910119895)(119882
120573
(119909119894)(119910119895)) is given by
119901(119909119894)(119910119895)
= 1 (119894 = 119895) 119901
120573
119872(119909 119910) exp[
[
minus120573 sum
119895d(119894119895)=1
120582119894119895]
]
+ sum
119896ge1
sum
1198970=1198941198971119897119896119897119896+1
=119895
prod
0le119904le119896
1 (d (119897119904 119897119904+1
) = 1)
times 120582119897119904119897119904+1
int
120573
0
d120591119904exp[
[
minus (120591119904+1
minus 120591119904) sum
119895d(119897119904119895)=1
120582119897119904119895]
]
times 1 (0 = 1205910lt 120591
1lt sdot sdot sdot lt 120591
119896lt 120591
119896+1= 120573)
(41)
where 119901120573
119872(119909 119910) denotes the transition probability density for
the Brownian motion to pass from 119909 to 119910 on119872 in time 120573
119901
120573
119872(119909 119910) =
1
(2120587120573)
1198892
times sum
119899=(1198991119899119889)isinZ119889
exp(minus
1003816100381610038161003816
119909 minus 119910 + 119899
1003816100381610038161003816
2
2120573
)
(42)
In view of (13) the quantity 119901(119909119894)(119910119895)
and its derivatives areuniformly bounded
119901(119909119894)(119910119895)
10038161003816100381610038161003816
nabla119909119901(119909119894)(119910119895)
10038161003816100381610038161003816
10038161003816100381610038161003816
nabla119910119901(119909119894)(119910119895)
10038161003816100381610038161003816
le 119901119872
119909 119910 isin 119872 119894 119895 isin Γ
(43)
where 119901119872
= 119901119872(120573) isin (0 +infin) is a constant
We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future
Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations overΛ with ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a pairing between xlowast
Λand ylowast
Λ
Then Plowast
xlowastΛylowastΛ120574denotes the product measure on119882
120573
xlowastΛylowastΛ120574
P120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)P120573
(119909119894)120574(119909119894) (44)
Furthermore P120573
xlowastΛylowastΛ
stands for the sum measure on119882
120573
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
= sum
120574
P120573
xlowastΛylowastΛ120574 (45)
According to Definition 10 under the measure P120573
xlowastΛylowastΛ120574
the trajectories 120596119909119894
isin 119882
120573
(119909119894)120574(119909119894)constituting 120596
Λare inde-
pendent components (Here the term independence is usedin the measure-theoretical sense)
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Mathematical Physics
defined as a collection of pairs [(119909 119894) (119910 119895)]120574 with 119894 119895 isin Λ
119909 isin xlowast(119894) and 119910 isin ylowast(119895) with the properties that (i) forall119894 isin Λ
and 119909 isin xlowast(119894) there exist unique 119895 isin Λ and 119910 isin ylowast(119895)such that (119909 119894) and (119910 119895) form a pair and (ii) forall119895 isin Λ and119910 isin ylowast(119895) there exist unique 119894 isin Λ and 119909 isin xlowast(119894) suchthat (119909 119894) and (119910 119895) form a pair (Owing to the condition♯xlowast
Λ= ♯ylowast
Λ these properties are equivalent) It is convenient
to write [(119909 119894) (119910 119895)]120574= [(119909 119894) 120574(119909 119894)]
Next consider the Cartesian product
119882
120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
120573
(119909119894)120574(119909119894) (38)
and the disjoint union
119882
120573
xlowastΛylowastΛ
= ⋃
120574
119882
120573
xlowastΛylowastΛ120574 (39)
Accordingly an element 120596Λ
isin 119882
120573
xlowastΛylowastΛ120574in (38) represents a
collection of paths 120596119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 120573
starting at (119909 119894) and ending up at 120574(119909 119894) We say that 120596Λis a
path configuration in (or over) Λ
The presence of matchings in the above construction is afeature of the bosonic nature of the systems under consider-ation
We will work with standard sigma algebras (generated bycylinder sets) in119882
120573
(119909119894)(119910119895)119882120573
xlowastΛylowastΛ120574 and119882
120573
xlowastΛylowastΛ
Definition 9 In what follows 120585(120591) 120591 ge 0 stands for theMarkov process on 119872 times Γ with cadlag trajectories deter-mined by the generator L acting on a function (119909 119894) isin 119872 times
Λ 997891rarr 120595(119909 119894) by
L120595 (119909 119894) = minus
1
2
Δ120595 (119909 119894)
+ sum
119895d(119894119895)=1
120582119894119895int
119872
V (d119910) [120595 (119910 119895) minus 120595 (119909 119894)]
(40)
In the probabilistic literature such processes are referred toas Levy processes see for example [14]
Pictorially a trajectory of process 120585 moves along 119872
according to the Brownian motion with the generator minusΔ2and changes the index 119894 isin Γ from time to time in accor-dance with jumps occurring in a Poisson process of ratesum
119895d(119894119895)=1 120582119894119895 In other words while following a Brownian
motion rule on 119872 having index 119894 isin Γ and being at point119909 isin 119872 themoving particle experiences an urge to jump from119894 to a neighboring vertex 119895 and to a point 119910 at rate 120582
119894119895V(d119910)
After a jump the particle continues the Brownian motion on119872 from 119910 and keeps its new index 119895 until the next jump andso on
For a given pairs (119909 119894) (119910 119895) isin 119872 times Γ we denote byP120573
(119909119894)(119910119895)the nonnormalised measure on 119882
120573
(119909119894)(119910119895)induced
by 120585 That is under measure P120573
(119909119894)(119910119895)the trajectory at time
120591 = 0 starts from the point 119909 and has the initial index 119894 whileat time 120591 = 120573 it is at the point 119910 and has the index 119895The value119901(119909119894)(119910119895)
= P120573
(119909119894)(119910119895)(119882
120573
(119909119894)(119910119895)) is given by
119901(119909119894)(119910119895)
= 1 (119894 = 119895) 119901
120573
119872(119909 119910) exp[
[
minus120573 sum
119895d(119894119895)=1
120582119894119895]
]
+ sum
119896ge1
sum
1198970=1198941198971119897119896119897119896+1
=119895
prod
0le119904le119896
1 (d (119897119904 119897119904+1
) = 1)
times 120582119897119904119897119904+1
int
120573
0
d120591119904exp[
[
minus (120591119904+1
minus 120591119904) sum
119895d(119897119904119895)=1
120582119897119904119895]
]
times 1 (0 = 1205910lt 120591
1lt sdot sdot sdot lt 120591
119896lt 120591
119896+1= 120573)
(41)
where 119901120573
119872(119909 119910) denotes the transition probability density for
the Brownian motion to pass from 119909 to 119910 on119872 in time 120573
119901
120573
119872(119909 119910) =
1
(2120587120573)
1198892
times sum
119899=(1198991119899119889)isinZ119889
exp(minus
1003816100381610038161003816
119909 minus 119910 + 119899
1003816100381610038161003816
2
2120573
)
(42)
In view of (13) the quantity 119901(119909119894)(119910119895)
and its derivatives areuniformly bounded
119901(119909119894)(119910119895)
10038161003816100381610038161003816
nabla119909119901(119909119894)(119910119895)
10038161003816100381610038161003816
10038161003816100381610038161003816
nabla119910119901(119909119894)(119910119895)
10038161003816100381610038161003816
le 119901119872
119909 119910 isin 119872 119894 119895 isin Γ
(43)
where 119901119872
= 119901119872(120573) isin (0 +infin) is a constant
We suggest a term ldquonon-normalised Brownian bridgewith jumpsrdquo for themeasure but expect that a better termwillbe proposed in future
Definition 10 Suppose that xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ andylowastΛ= ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations overΛ with ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a pairing between xlowast
Λand ylowast
Λ
Then Plowast
xlowastΛylowastΛ120574denotes the product measure on119882
120573
xlowastΛylowastΛ120574
P120573
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)P120573
(119909119894)120574(119909119894) (44)
Furthermore P120573
xlowastΛylowastΛ
stands for the sum measure on119882
120573
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
= sum
120574
P120573
xlowastΛylowastΛ120574 (45)
According to Definition 10 under the measure P120573
xlowastΛylowastΛ120574
the trajectories 120596119909119894
isin 119882
120573
(119909119894)120574(119909119894)constituting 120596
Λare inde-
pendent components (Here the term independence is usedin the measure-theoretical sense)
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 7
As in [1] we will work with functionals on119882
120573
xlowastΛylowastΛ120574repre-
senting integrals along trajectories The first such functionalhΛ(120596
Λ) is given by
hΛ (120596Λ) = sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (120596
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 120596
11990910158401198941015840)
(46)
Here introducing the notation 119906119909119894(120591) = 119906(120596
119909119894 120591) and
11990611990910158401198941015840(120591) = 119906(120596
11990910158401198941015840 120591) for the positions in 119872 of paths 120596
119909119894isin
119882
120573
(119909119894)120574(119909119894)and 120596
11990910158401198941015840 isin 119882
120573
(11990910158401198941015840)120574(11990910158401198941015840)at time 120591 we define
h119909119894 (120596119909119894) = int
120573
0
d120591119880(1)(119906
119894119909(120591)) (47)
Next with 119897119909119894(120591) and 119897
11990910158401198941015840(120591) standing for the indices of 120596
119909119894
and 12059611990910158401198941015840 at time 120591
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 120596
11990910158401198941015840)
= int
120573
0
d120591[
[
sum
1198951015840isinΓ
119880
(2)(119906
119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119897
11990910158401198941015840 (120591))
+
1
2
sum
(119895101584011989510158401015840)isinΓtimesΓ
119869 (d (1198951015840 119895
10158401015840))
times 119881 (119906119894119909(120591) 119906
11989410158401199091015840 (120591))
times 1 (119897119909119894(120591) = 119895
1015840= 119895
10158401015840= 119897
11990910158401198941015840 (120591))
]
]
(48)
Next consider the functional hΛ(120596Λ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ As before we assume that ♯xlowast(119895) le 120581Define
hΛ (120596Λ| xlowast
ΓΛ) = hΛ (120596
Λ) + hΛ (120596
Λ|| xlowast
ΓΛ) (49)
Here hΛ(120596Λ) is as in (46) and
hΛ (120596Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)
times (120596119909119894 (119909
1015840 119894
1015840))
(50)
where in turn
h(119909119894)(1199091015840
1198941015840
)(120596
119909119894 (119909
1015840 119894
1015840))
= int
120573
0
d120591 [119880(2)(119906
119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119894
1015840)
+ sum
119895isinΓ119895 = 1198941015840
119869 (d (119895 1198941015840))
times119881 (119906119894119909(120591) 119909
1015840) 1 (119897
119909119894(120591) = 119895)]
(51)
The functionals hΛ(120596Λ) and hΛ(120596
Λ|| xlowast
ΓΛ) are interpreted as
energies of path configurations Compare (214) and (238)
in [1]Finally we introduce the indicator functional 120572
Λ(120596
Λ)
120572Λ(120596
Λ) =
1 if index 119897119909119894(120591) isin Λ
forall120591 isin [0 120573] 119894 isin Λ 119909 isin xlowast (119894) 0 otherwise
(52)
It can be derived from known results [11 15ndash17] (for a directargument see [18]) that the following assertion holds true
Lemma 11 For all 119911 120573 gt 0 and a finite Λ the Gibbs operatorsG
Λand G
Λ|xlowastΓΛ
act as integral operators inH(Λ)
(GΛ120601) (xlowast
Λ) = int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)
V (d119910)KΛ(xlowast
Λ ylowast
Λ)120601 (ylowast
Λ)
(GΛ|xlowastΓΛ
120601) (xlowastΛ)
= int
119872lowastΛ
prod
119895isinΛ
prod
119910isinylowast(119895)V (d119910)K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)120601 (ylowast
Λ)
(53)
Moreover the integral kernels KΛ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ|
xlowastΓΛ
) vanish if ♯xlowastΛ
= ♯ylowastΛ On the other hand when ♯xlowast
Λ=
♯ylowastΛ the kernels K
Λ(xlowast
Λ ylowast
Λ) and K
Λ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ) admit the
following representations
KΛ(xlowast
Λ ylowast
Λ) = 119911
♯xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ)]
(54)
KΛ(xlowast
Λ ylowast
Λ| xlowast
ΓΛ)
= 119911
♯ xlowastΛ
int
119882lowast
xlowastΛylowastΛ
P120573
xlowastΛylowastΛ
(d120596Λ) 120572
Λ(120596
Λ)
times exp [minushΛ (120596Λ| xlowast
ΓΛ)]
(55)
The ingredients of these representations are determined in (46)ndash(51)
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
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Stochastic AnalysisInternational Journal of
8 Advances in Mathematical Physics
Remark 12 As before we stress that owing to (16) and (17)a nonzero contribution to the integral in the RHS of (54) canonly come from a path configuration 120596
Λ= 120596
119909119894 such that
forall120591 isin [0 120573] and forall119895 isin Γ the number of paths 120596119909119894
with index119897119909119894(120591) = 119895 is less than or equal to 120581 Likewise the integral in
the RHS of (55) receives a non-zero contribution only fromconfigurations120596
Λ= 120596
lowast
119909119894 such that forall site 119895 isin Γ the number
of paths 120596119909119894
with index 119897119909119894(120591) = 119895 plus the cardinality ♯xlowast(119895)
does not exceed 120581
22 The Representation for the Partition Function The FKrepresentations of the partition functions Ξ(Λ) = Ξ
120573119911(Λ)
in (27) and Ξ(Λ | xlowastΓΛ
) in (146) reflect a specific characterof the traces trG
Λand trG
Λ|xlowastΓΛ
in H(Λ) The source of acomplication here is the jump terms in the Hamiltonians119867
Λ
and119867Λ|xlowastΓΛ
in (11) and (12) respectively In particular we willhave to pass from trajectories of fixed time-length 120573 to loopsof a variable time length To this end a given matching 120574
is decomposed into a product of cycles and the trajectoriesassociated with a given cycle are merged into closed paths(loops) of a time-lengthmultiple of120573 (A similar constructionhas been performed in [18])
To simplify the notation we omit wherever possible theindex 120573
Definition 13 For given (119909 119894) (119910 119895) isin 119872 times Γ the symbol119882
lowast
(119909119894)(119910119895)denotes the disjoint union
119882
lowast
(119909119894)(119910119895)= ⋃
119896=01
119882
119896120573
(119909119894)(119910119895) (56)
In other words119882lowast
(119909119894)(119910119895)is the space of pathsΩlowast
= Ω
lowast
(119909119894)(119910119895)
in119872timesΓ of a variable time-length 119896120573 where 119896 = 119896(Ω
lowast
) takesvalues 1 2 and called the lengthmultiplicity with the end-points (119909 119894) and (119910 119895)The formal definition follows the sameline as in (37) and we again use the notation Ω
lowast
(120591) and thenotation (119906(Ω
lowast
120591) 119897(Ω
lowast
120591)) for the pair of the position andthe index of path Ω
lowast at time 120591 Next we call the particleconfiguration Ω
lowast
(120591 + 120573119898) 0 le 119898 lt 119896(Ω
lowast
) the temporalsection (or simply the section) of Ωlowast at time 120591 isin [0 120573] Wealso callΩlowast
(119909119894)(119910119895)isin 119882
lowast
(119909119894)(119910119895)a path (from (119909 119894) to (119910 119895))
A particular role will be played by closed paths (loops)with coinciding endpoints (where (119909 119894) = (119910 119895)) Accord-ingly we denote by 119882
lowast
119909119894the set 119882lowast
(119909119894)(119909119894) An element of
119882
lowast
119909119894is denoted by Ω
lowast
119909119894or in short by Ω
lowast and called a loopat vertex 119894 (The upper index lowast indicates that the lengthmultiplicity is unrestricted) The length multiplicity of a loopΩ
lowast
119909119894isin 119882
lowast
119909119894is denoted by 119896(Ωlowast
119909119894) or 119896
119909119894 It is instructive to
note that as topological object a given loop Ωlowast admits amultiple choice of the initial pair (119909 119894) it can be representedby any pair (119906(Ωlowast
120591) 119897(Ω
lowast 120591)) at a time 120591 = 119897120573 where 119897 =
1 119896(Ωlowast) As above we use the term the temporal sectionat time 120591 isin [0 120573] for the particle configuration Ω
lowast
119909119894(120591 +
120573119898) 0 le 119898 lt 119896119909119894 and employ the alternative notation
(119906(120591 + 120573119898Ω
lowast) 119897(120591 + 120573119898Ω
lowast)) addressing the position and
the index ofΩlowast at time 120591 + 120573119898 isin [0 120573119896(Ω
lowast)]
Definition 14 Suppose xlowastΛ= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ are particle configurations over Λwith ♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ We
consider the Cartesian product
119882
lowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
(119909119894)120574(119909119894) (57)
and the disjoint union
119882
lowast
xlowastΛylowastΛ
= ⋃
120574
119882
lowast
xlowastΛylowastΛ120574 (58)
Accordingly an element ΩlowastΛisin 119882
lowast
xlowastΛylowastΛ120574in (58) represents a
collection of paths Ωlowast119909119894 119909 isin xlowast(119894) 119894 isin Λ of time-length 119896120573
starting at (119909 119894) and ending up at (119910 119895) = 120574(119909 119894) We say thatΩ
lowast
Λisin 119882
lowast
xlowastΛylowastΛ
is a path configuration in (or over) ΛAgain loops play a special role and deserve a particular
notation Namely119882lowast
xlowastΛ
denotes the Cartesian product
119882
lowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)119882
lowast
119909119894 (59)
and 119882
lowast
Λstands for the disjoint union (or equivalently the
Cartesian power)
119882
lowast
Λ= ⋃
xlowastΛisin119872lowastΛ
119882
lowast
xlowastΛ
= times
119894isinΛ
119882
lowast
119894
where 119882
lowast
119894= ⋃
xlowastisin119872lowast( times
119909isinxlowast119882
lowast
119909119894)
(60)
Denote by Ωlowast = Ωlowast(119894) 119894 isin Λ isin 119882
lowast
Λa collection of
loop configurations at vertices 119894 isin Λ starting and ending upat particle configurations xlowast(119894) isin 119872
lowast (note that some of theΩlowast(119894)rsquos may be empty) The temporal section (or in shortthe section) Ωlowast(120591) of Ωlowast at time 120591 is defined as the particleconfiguration formed by the pointsΩlowast
119909119894(120591+120573119898)where 119894 isin Λ
119909 isin xlowast(119894) and 0 le 119898 lt 119896119909119894
As before consider the standard sigma algebras of subsetsin the spaces 119882lowast
(119909119894)(119910119895) 119882
119909119894 119882lowast
xlowastΛylowastΛ120574 119882lowast
xlowastΛylowastΛ
119882lowast
xlowastΛ
and 119882
lowast
Λ
introduced in Definitions 13 and 14 In particular the sigmaalgebra of subsets in119882lowast
Λwill be denoted by W
Λ we comment
on some of its specific properties in Section 31 (An infinite-volume version119882
lowast
Γof119882lowast
Λis treated in Section 32 and after)
Definition 15 Given points (119909 119894) (119910 119895) isin 119872 times Γ we denoteby Plowast
(119909119894)(119910119895)the sum measure on119882
lowast
(119909119894)(119910119895)
Plowast
(119909119894)(119910119895)= sum
119896=01
P119896120573
(119909119894)(119910119895) (61)
Further Plowast
119909119894denotes the similar measure on119882
lowast
119909119894
Plowast
119909119894= sum
119896=01
P119896120573
119909119894 (62)
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 9
Definition 16 Let xlowastΛ
= xlowast(119894) 119894 isin Λ isin 119872
lowastΛ and ylowastΛ
=
ylowast(119895) 119895 isin Λ isin 119872
lowastΛ be particle configurations overΛ with♯xlowast
Λ= ♯ylowast
Λ Let 120574 be a matching between xlowast
Λand ylowast
Λ and we
define the product measure Plowast
xlowastΛylowastΛ120574
Plowast
xlowastΛylowastΛ120574= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
(119909119894)120574(119909119894) (63)
and the sum measure
Plowast
xlowastΛylowastΛ
= sum
120574
Plowast
xlowastΛylowastΛ120574 (64)
Next symbolPlowast
xlowastΛ
stands for the product measure on119882lowast
xlowastΛ
Plowast
xlowastΛ
= times
119894isinΛ
times
119909isinxlowast(119894)Plowast
119909119894 (65)
Finally dΩlowastΛyields the measure on119882
lowast
Λ
dΩlowastΛ= dxlowast
Λtimes P
lowast
xlowastΛ
(dΩlowastΛ) (66)
Here for xlowastΛ
= xlowast(119894) 119894 isin Λ we set dxlowastΛ
=
prod119894isinΛ
prod119909isinxlowast(119894)V(d119909) For sites 119894 with xlowast(119894) = 0 the corre-
sponding factors are trivial measures sitting on the emptyconfigurations
We again need to introduce energy-type functionalsrepresented by integrals along loops More precisely wedefine the functionals hΛ(Ωlowast
Λ) and hΛ(Ωlowast
Λ| xlowast
ΓΛ) which are
modifications of the above functionals hΛ(120596Λ) and hΛ(120596
Λ|
xlowastΓΛ
) confer (46) and (50) Say for a loop configurationΩlowastΛ=
Ω
lowast
119909119894 over Λ with an initial and final particle configuration
xlowastΛ= xlowast(119894) 119894 isin Λ
hΛ (Ωlowast
Λ)
= sum
119894isinΛ
sum
119909isinxlowast(119894)h119909119894 (Ωlowast
119909119894)
+
1
2
sum
(1198941198941015840)isinΛtimesΛ
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
times 1 ((119909 119894) = (119909
1015840 119894
1015840)) h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
x119894 Ωlowast
11990910158401198941015840)
(67)
To determine the functionals h119909119894(Ωlowast
119909119894) and h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894
Ω
lowast
11990910158401198941015840) we set for given 119898 = 0 1 119896
119909119894minus 1 and 119898
1015840=
0 1 11989611990910158401198941015840 minus 1
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
119909119894)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
119909119894)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(68)
A (slightly) shortened notation 119897119894119909(120591 + 120573119898) is used for the
index 119897119909119894(120591 + 120573119898Ω
lowast
119909119894) and 119906
119894119909(120591 + 120573119898) for the position
119906(120591 + 120573119898Ω
lowast
119909119894) for Ωlowast
119909119894(120591) isin 119872 times Γ of the section Ω
lowast
119909119894(120591)
of the loop Ω
lowast
119909119894at time 120591 and similarly with 119897
11989410158401199091015840(120591 + 120573119898
1015840)
and 11990611989410158401199091015840(120591 + 120573119898
1015840) (Note that the pairs (119909 119894) and (119909
1015840 119894
1015840)may
coincide) Then
h(119909119894) (Ωlowast
119909119894)
= int
120573
0
d120591[
[
sum
0le119898lt119896119909119894
119880
(1)(119906
119909119894(120591 + 120573119898))
+ sum
0le119898lt1198981015840lt119896119909119894
sum
119895isinΓ
1 (119897119909119894(120591 + 120573119898)
= 119895 = 119897119909119894(120591 + 120573119898
1015840))
times119880
(2)(119906
119909119894(120591 + 120573119898) 119906
119909119894(120591 + 120573119898
1015840))
]
]
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840 )
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(69)
Next the functional119861(ΩlowastΛ) takes into account the bosonic
character of the model
119861 (Ωlowast
Λ) = prod
119894isinΛ
prod
119909isinxlowast(119894)
119911
119896119909119894
119896119909119894
(70)
The factor 119896minus1119909119894
in (70) reflects the fact that the starting pointof a loop Ω
lowast
119909119894may be selected among points 119906(120573119898Ω
lowast
119909119894)
arbitrarily
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Advances in Mathematical Physics
Next we define the functional hΛ(ΩlowastΛ| xlowast
ΓΛ) for xlowast
ΓΛ=
xlowast(119895) 119895 isin Γ Λ again assuming that ♯xlowast(119895) le 120581 Set
hΛ (Ωlowast
Λ| xlowast
ΓΛ) = hΛ (Ω
lowast
Λ) + hΛ (Ω
lowast
Λ|| xlowast
ΓΛ) (71)
Here hΛ(ΩlowastΛ) is as in (67) and
hΛ (Ωlowast
Λ|| xlowast
ΓΛ)
= sum
(1198941198941015840)isinΛtimes(ΓΛ)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(72)
where in turn
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591
times [119880
(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(73)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ) have a
natural interpretation as energies of loop configurationsFinally as before the functional 120572
Λ(Ωlowast
Λ) is the indicator
that the collection of loopsΩlowastΛ= Ω
lowast
119909119894 does not quit Λ
120572Λ(Ω
lowast
Λ) =
1 if Ωlowast
119909119894(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896119909119894
0 otherwise(74)
Like above we invoke known results [11 15ndash17] to estab-lish the following statement (again a direct argument can befound in [18])
Lemma 17 For all finite Λ sub Γ and 119911 120573 gt 0 satisfying (22)the partition functions Ξ(Λ) in (27) and Ξ(Λ | xlowast
ΓΛ) in (27)
and (28) admit the representations as converging integrals
Ξ (Λ) = int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ)] (75)
Ξ (Λ | xlowastΓΛ
)
= int
119882lowast
Λ
dΩlowastΛ119861 (Ω
lowast
Λ) 120572
Λ(Ω
lowast
Λ) exp [minushΛ (Ω
lowast
Λ|| xlowast
ΓΛ)]
(76)
with the ingredients introduced in (60)ndash(74)
Again we emphasis that the non-zero contribution tothe integral in (75) can only come from loop configurationsΩlowast
Λ= Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Λ such that forall vertex 119895 isin Λ and
120591 isin [0 120573] the total number of pairs (119906119909119894(120591+119898120573) 119897
119909119894(120591+119898120573))
with 0 le 119898 lt 119896119909119894 and 119897(120591 + 119898120573) = 119895 does not exceed 120581
Remark 18 The integrals in (75) and (76) represent examplesof partition functions which will be encountered in the forth-coming sections See (96) (97) (101) (103) (105) (107) (111)and (113) below A general form of such a partition functiontreated as an integral over a set of loop configurations ratherthan a trace in a Hilbert space is given in (96) and (97)
23 The Representation for the RDM Kernels Let Λ0 Λ be
finite sets Λ0sub Λ sub Γ The construction developed in
Section 22 also allows us towrite a convenient representationfor the integral kernels of the RDMsRΛ
0
Λ(see (31)) andRΛ
0
Λ|xlowastΓΛ
In accordancewith Lemma 11 and the definition ofRΛ0
Λin (31)
the operator RΛ0
Λacts as an integral operator inH(Λ
0)
(RΛ0
Λ120601) (xlowast0)
= int
119872lowastΛ0
prod
119895isinΛ0
prod
119910isinylowast(119895)V (d119910) FΛ
0
Λ(xlowast0 ylowast0)120601 (ylowast0)
(77)
where
FΛ0
Λ(xlowast0 ylowast0)
= int
119872lowastΛΛ0
prod
119895isinΛΛ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ(xlowast0 or zlowast
ΛΛ0 ylowast0 or zlowast
ΛΛ0)
=
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
Ξ (Λ)
(78)
We employ here and below the notation xlowast0 and ylowast0 forparticle configurations xlowast
Λ0 = xlowast(119894) 119894 isin Λ
0 and ylowast
Λ0 =
ylowast(119895) 119895 isin Λ
0 overΛ0 Next xlowast0orzlowast
ΛΛ0 ylowast0orzlowast
ΛΛ0 denotes
the concatenated configurations over ΛSimilarly the RDM RΛ
0
Λ|xlowastΓΛ
is determined by its integral
kernel FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) again admitting the representation
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) =ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
Ξ (Λ)
(79)
As in [1] we call FΛ0
Λand FΛ
0
Λ|xlowastΓΛ
the RDM kernels (inshort RDMKs) The focus of our interest is the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in
(78) and (79) To introduce the appropriate representation forthese quantities we need some additional definitions
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 11
Definition 19 Repeating (57)-(58) symbol 119882lowast
xlowast0 ylowast0 denotesthe disjoint union ⋃
1205740119882
lowast
xlowast0 ylowast01205740 over matchings 120574
0
between xlowast0 and ylowast0 Accordingly elementΩlowast0 = Ωlowastxlowast0 ylowast01205740 isin119882
lowast
xlowast0 ylowast01205740 yields a collection of pathsΩlowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)1205740(119909119894)
lying in 119872 times Γ Each path Ω
lowast
(119909119894)1205740(119909119894)
has time lengths120573119896
(119909119894)(119910119895) begins at (119909 119894) and ends up at (119910 119895) = 120574
0(119909 119894)
where 119909 isin xlowast(119894) 119910 isin ylowast(119895) Like above we will use for Ωlowast0
the term a path configuration over Λ0 Repeating (63)-(64)we obtain the measures Plowast
xlowast0 ylowast01205740 on 119882
lowast
xlowast0 ylowast01205740 and Plowast
xlowast0ylowast0
on119882
lowast
xlowast0 ylowast0
The assertion of Lemma 20 below again follows directlyfrom known results in conjunction with calculations of thepartial trace trH
ΛΛ0
in HΛ The meaning of new ingredients
in (79)ndash(82) is explained below
Lemma 20 The quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) emerging in
(78) is set to be 0 when ♯xlowast0 = ♯ylowast0 On the other hand for♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
)]ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
(80)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0)
times 120572Λ(Ω
lowast
ΛΛ0) 1 (Ωlowast
ΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(81)
Similarly the quantity ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) from (79)
vanishes when ♯xlowast0 = ♯ylowast0 For ♯xlowast0 = ♯ylowast0
ΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ)
= int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
) 119861 (Ωlowast0
)
times 120572Λ(Ω
lowast0
) 1 (Ωlowast0 isin FΛ0
)
times exp [minushΛ0
(Ωlowast0
| xlowastΓΛ
)]
timesΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
(82)
where
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
)
= int
119882lowast
ΛΛ0
dΩlowastΛΛ0119861 (Ω
lowast
ΛΛ0) 120572Λ
(Ωlowast
ΛΛ0)
times 1 (ΩlowastΛΛ0 isin F
Λ0
)
times exp [minushΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)]
(83)
These representations hold forall119911 120573 gt 0 and finite Λ0sub Λ sub Γ
Let us define the functionals 119861(Ωlowast0) 120572Λ(Ω
lowast0
) 1(Ωlowast0 isin
FΛ0
) 1(ΩlowastΛΛ0 isin FΛ
0
)hΛ0
(Ωlowast0
) hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)hΛ0
(Ωlowast0xlowast
ΓΛ) and hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (79)ndash(83)(The functionals 119861(Ωlowast
ΛΛ0) and 120572
Λ(Ωlowast
ΛΛ0) are defined as (70)
and (74) respectively replacing Λ with Λ Λ
0)To this end let Ωlowast0 = Ω
lowast
xlowast0ylowast01205740 isin 119882
lowast
xlowast0 ylowast01205740 be apath configuration represented by a collection of pathsΩ
lowast
(119909119894)1205740(119909119894)
isin 119882
lowast
(119909119894)120574lowast0(119909119894)
(Ωlowast
119909119894in short) with end points (119909 119894)
and (119910 119895) = 120574
0(119909 119894) of time-length120573119896
(119909119894)(119910119895)The functional
119861(Ωlowast0
) is given by
119861 (Ωlowast0
) = prod
119894isinΛ
prod
119909isinxlowast(119894)119911
119896(119909119894)(119910119895)
(84)
The functional 120572Λ(Ω
lowast0
) is again an indicator
120572Λ(Ω
lowast0
) =
1 if Ωlowast
(119909119894)1205740(119909119894)
(120591) isin 119872 times Λ
forall119894 isin Λ 119909 isin xlowast (119894) 0 le 120591 le 120573119896
(119909119894)(119910119895)
0 otherwise
(85)
Now let us define the indicator function 1(sdot isin FΛ0
) in(80)ndash(83) The factor 1(Ωlowast0 isin FΛ
0
) equals one if and onlyif every path Ω
lowast
(119909119894)(119910119895)from Ωlowast0 of time-length 120573119896
(119909119894)(119910119895)
starting at (119909 119894) isin 119872timesΛ
0 and ending up at (119910 119895) = 120574
0(119909 119894) isin
119872timesΛ
0 remains in119872times (Λ Λ
0) at the intermediate times 120573119897
for 119897 = 1 119896(119909119894)(119910119895)
minus 1
Ω
lowast
(119909119894)(119910119895)(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
(119909119894)(119910119895)minus 1 (86)
(when 119896(119909119894)(119910119895)
= 1 this is not a restriction)Furthermore suppose that Ωlowast
ΛΛ0 = Ωlowastxlowast
ΛΛ0
is a loop
configuration over Λ Λ
0 with the initialend configurationxlowastΛΛ0 = xlowast(119894) 119894 isin Λ Λ
0 represented by a collection of
loops Ωlowast
119909119894 119894 isin Λ Λ
0 119909 isin xlowast(119894) Then 1(Ωlowast
ΛΛ0 isin FΛ
0
) = 1
if and only if each loop Ω
lowast
119909119894of time-length 120573119896
119909119894 beginning
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
12 Advances in Mathematical Physics
and finishing at (119909 119894) isin 119872 times (Λ Λ
0) does not enter the set
119872times Λ
0 at times 120573119897 for 119897 = 1 119896119909119894
minus 1
Ω
lowast
119909119894(119897120573) notin 119872 times Λ
0 forall119897 = 1 119896
119909119894minus 1 (87)
(again if 119896119909119894
= 1 this is not a restriction)The functional hΛ
0
(Ωlowast0
) in (80) gives the energy of thepath configuration Ωlowast0 and is introduced similarly to (67)mutatis mutandis Next the functional hΛΛ
0
(ΩlowastΛΛ0 | Ω
lowast0
) in(81) represents the energy of the loop configurationΩlowast
ΛΛ0 in
the potential field generated by the path configurationΩlowast0
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
) = hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast0 || Ωlowast
ΛΛ0)
(88)
Here the summand hΛΛ0
(ΩlowastΛΛ0) yields the energy of the
loop configurationΩlowastΛΛ0 again confer (67) Further the term
h(Ωlowast0 || ΩlowastΛΛ0) yields the energy of interaction betweenΩ
lowast0
and ΩlowastΛΛ0 for a pathloop configurations Ωlowast0 = Ω
lowast
119909119894 isin
119882
lowast
xlowast0 ylowast01205740 and aΩlowastΛΛ0 = Ω
lowast
11990910158401198941015840 isin 119882
lowast
ΛΛ0 we set
h (Ωlowast0 || ΩlowastΛΛ0)
= sum
(1198941198941015840
)isinΛ0times(ΛΛ
0
)
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
(89)
Here for a pathΩlowast
119909119894= Ω
lowast
(119909119894)1205740(119909119894)
of time-length 120573119896(119909119894)120574
0(119909119894)
and a loopΩ
lowast
11990910158401198941015840 of time-length 120573119896
119909119894
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896(119909119894)1205740(119909119894)
sum
0le1198981015840lt11989611990910158401198941015840
int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840))
times 1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 11989711990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
1198951198951015840isinΓtimesΓ
119869 (d (119895 1198951015840))119881 (119906
119894119909(120591 + 120573119898) 119906
11989410158401199091015840 (120591 + 120573119898
1015840))
times1 (119897(119909119894)120574
0(119909119894)
(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
(90)
Here in turn we employ the shortened notation for thepositions and indices of the sections Ωlowast
(119909119894)1205740(119909119894)
(120591 + 120573119898) andΩ
lowast
11990910158401198941015840(120591 + 120573119898
1015840) of Ωlowast
(119909119894)1205740(119909119894)
and Ω
lowast
11990910158401198941015840 at times 120591 + 120573119898 and
120591 + 120573119898
1015840 respectively
119906119894119909(120591 + 120573119898) = 119906 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
119897119894119909(120591 + 120573119898) = 119897 (120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)
11990611989410158401199091015840 (120591 + 120573119898
1015840) = 119906 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
11989711989410158401199091015840 (120591 + 120573119898
1015840) = 119897 (120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840)
(91)
Further the functional hΛ0
(Ωlowast0
| xlowastΓΛ
) in (82) isdetermined as in (71)ndash(73) with Ω
lowast
(119909119894)1205740(119909119894)
instead of Ωlowast
119909119894
Next for hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
or xlowastΓΛ
) in (83) we set
hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
or xlowastΓΛ
)
= hΛΛ0
(Ωlowast
ΛΛ0) + h (Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
(92)
Here again the summand hΛΛ0
(ΩlowastΛΛ0) is determined as in
(67) Next the term hΛΛ0
(ΩlowastΛΛ0 || Ω
lowast0
or xlowastΓΛ
) is definedsimilarly to (72)-(73)
hΛΛ0
(Ωlowast
ΛΛ0 || Ω
lowast0
or xlowastΓΛ
)
= sum
(1198941198941015840)isin(ΛΛ
0
)timesΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
+ sum
119894isinΛΛ0
sum
119909isinxlowast(119894)1199091015840isinxlowast(1198941015840)h(119909119894)(119909
1015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
(93)
with
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 Ω
lowast
11990910158401198941015840)
= sum
0le119898lt119896119909119894
sum
0le1198981015840lt119896(11990910158401198941015840)1205740(11990910158401198941015840)
times int
120573
0
d120591
times[
[
sum
119895isinΓ
119880
(2)(119906
119909119894(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times 1 (119897119909119894(120591 + 120573119898) = 119895 = 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
+ sum
(1198951198951015840
)isinΓtimesΓ
119869 (d (119895 1198951015840))119881
times (119906119894119909(120591 + 120573119898) 119906
11990910158401198941015840 (120591 + 120573119898
1015840))
times1 (119897119909119894(120591 + 120573119898) = 119895 = 119895
1015840= 119897
11990910158401198941015840 (120591 + 120573119898
1015840))
]
]
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 13
h(119909119894)(1199091015840
1198941015840
)(Ω
lowast
119909119894 (119909
1015840 119894
1015840))
= sum
0le119898lt119896119909119894
int
120573
0
d120591 [119880(2)(119906
119909119894(120591 + 120573119898) 119909
1015840)
times 1 (119897119909119894(120591 + 120573119898) = 119894
1015840)
+ sum
119895isinΓ
119869 (d (119895 1198941015840))119881 (119906
119894119909(120591 + 120573119898) 119909
1015840)
times1 (119897119909119894(120591 + 120573119898) = 119895) ]
(94)
As before the functionals hΛ(ΩlowastΛ) and hΛ(Ωlowast
Λ|| xlowast
ΓΛ)
have a natural interpretation as energies of loop configura-tions
Repeating the above observation non-zero contributionsto the integral in (80) come only from pairs (Ω
lowast0
ΩlowastΛΛ0)
such that forall119895 isin Γ and 120591 isin [0 120573] the total number of pairs(119906(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
) 119897(120591 + 120573119898Ω
lowast
(119909119894)1205740(119909119894)
)) with 0 le 119898 lt
119896(119909119894)120574
0(119909119894)
119894 isin Λ
0 and 119909 isin xlowast(119894) incident to the pathsof the configuration Ωlowast0 and pairs (119906(120591 + 120573119898
1015840 Ω
lowast
11990910158401198941015840) 119897(120591 +
120573119898
1015840 Ω
lowast
11990910158401198941015840)) with 0 le 119898
1015840lt 119896
(11990910158401198941015840) 1198941015840 isin Λ Λ
0 and 119909
1015840isin xlowast(1198941015840)
incident to the loops of the configuration ΩlowastΛΛ0 does not
exceed 120581 Similarly non-zero contributions to the integral in(82) come only from pairs (Ωlowast0Ωlowast
ΛΛ0) such that the above
inequality holdswhenwe additionally count points119909 isin xlowast(119895)The integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
) defined in (81) can beconsidered as a particular (although important) example ofa partition function in the volume Λ Λ
0 with a boundarycondition Ωlowast0 Note the presence of the subscript Λ indicat-ing that the loops contributing to Ξ
Λ0
Λ(ΛΛ
0| Ω
lowast0
) can jumpwithin volumeΛ only (owing to the indicator functional 120572
Λ)
On the other hand the presence of the indicator functional1(Ωlowast
ΛΛ0 isin FΛ
0
) in the integral (reflected in the upperscript
Λ
0 and the roof sign in the notation ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
))indicates a particular restriction on the jumps of the loopsforbidding them to visit setΛ0 at intermediate times 120573119898Thisis true also for the integral ΞΛ
0
Λ(Λ Λ
0| Ω
lowast0
or xlowastΓΛ
) in (83) itis a particular example of a partition function in the volumeΛ Λ
0 with a boundary conditionΩlowast0 or xlowastΓΛ
Other useful types of partition functions are Ξ
Γ0(Λ |
Ωlowast0
or ΩlowastΓ1) and ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1 or xlowast
Γ2) where the sets of
vertices Λ Λ0 Γ0 Γ1 and Γ
2 satisfy
Λ sub Γ
0sube Γ Γ
1 Γ
2sub Γ
Λ
Γ
1cap Γ
2= 0 Λ
0sub Γ (
Λ cup Γ
1cup Γ
2)
(95)
and ♯Λ ♯Λ
0lt +infin Accordingly Ωlowast0 is a (finite) configu-
ration over Λ0 ΩlowastΓ1 a (possibly infinite) loop configuration
over Γ1 and xlowastΓ2 a (possibly infinite) particle configuration
over Γ
2 The partition functions ΞΓ0(Λ | Ω
lowast0
or ΩlowastΓ1) and
ΞΓ0(Λ | Ω
lowast0
orΩlowastΓ1 or xlowast
Γ2) are given by
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1)]
(96)
ΞΓ0 (
Λ | Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)
= int
119882lowast
Λ
dΩlowastΛ120572Γ0 (Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| Ω
lowast0
orΩlowast
Γ1 or xlowast
Γ2)]
(97)
with the indicator 120572Γ0 as in (74) These partition functions
feature loop configurations Ωlowastformed by loops Ωlowast
119909119894 119894 isin
Λwhich start and finish in
Λ are confined to Γ
0 and movein a potential field generated by Ωlowast0 or Ωlowast
Γ1 where Ω
lowast0
=
Ω
lowast
(119909119894)1205740(119909119894)
andΩΓ1 = Ω
lowast
119909119894 119909 isin xlowast(119894) 119894 isin Γ
1 orΩlowast
Γ1 or xlowast
Γ2
where xlowastΓ2 = xlowast(119894) 119894 isin Γ
2 (The latter can be understood
as the concatenation of the loop configuration ΩlowastΓ1 over Γ1
and the loop configuration over Γ2 formed by the constanttrajectories sitting at points 119909 isin xlowast(119894) 119894 isin Γ
2) In (96) weassume that forall120591 isin [0 120573] and 119895 isin Γ the number
♯ (119909 119894 119898) 119894 isinΛ 119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
+ ♯ (119909 119894) 119894 isin Λ
0 119897 (120591 + 119898120573Ω
lowast
(119909119894)1205740(119909119894)
)
= 119895 0 le 119898 lt 119896(119909119894)120574
0(119909119894)
+ ♯ (119909 119894 119898) 119894 isin Γ
1
119897 (120591 + 119898120573Ω
lowast
119909119894) = 119895 0 le 119898 lt 119896
119909119894
(98)
does not exceed 120581 Analogously in (97) it is assumed that thesame is true for the above number plus the cardinality ♯xlowast(119895)
Such ldquomodifiedrdquo partition functions will be used in forth-coming sections
24 The FK-DLR Measure 120583Λin a Finite Volume The Gibbs
states 120593Λand 120593
Λ|xlowastΓΛ
give rise to probability measures 120583Λand
120583Λ|xlowastΓΛ
on the sigma algebraWΛof subsets of119882lowast
Λ The sigma
algebra WΛis constructed by following the structure of the
space 119882
lowast
Λ(a disjoint union of Cartesian products) confer
Definition 16 The measures 120583Λand 120583
Λ|xlowastΓΛ
are determined
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
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Stochastic AnalysisInternational Journal of
14 Advances in Mathematical Physics
by their Radon-Nykodym derivative 119901Λand 119901
Λ|xlowastΓΛ
relative tothe measure dΩlowast
Λ
119901Λ(Ω
lowast
Λ) =
120583Λ(dΩlowast
Λ)
dΩlowastΛ
=
1
Ξ (Λ)120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ)] Ω
lowast
Λisin 119882
lowast
Λ
119901Λ|xlowastΓΛ
(Ωlowast
Λ) =
120583Λ|xlowastΓΛ
(dΩlowastΛ)
dΩlowastΛ
1
Ξ (Λ | xlowastΓΛ
)
= 120572Λ(Ω
lowast
Λ) 119861 (Ω
lowast
Λ)
times exp [minushΛ (Ωlowast
Λ| xlowast
ΓΛ)] Ω
lowast
Λisin 119882
lowast
Λ
(99)
Given Λ
0sub Λ the sigma algebra W
Λ0 is naturally identified
with a sigma subalgebra of WΛ The restrictions of 120583
Λ
to WΛ0 and 120583
Λ|xlowastΓΛ
are denoted by 120583Λ0
Λand 120583Λ
0
Λ|xlowastΓΛ
thesemeasures are determined by their Radon-Nikodym deriva-tives 119901
Λ0
Λ(Ωlowast
Λ0) = 120583Λ
0
Λ(dΩlowast
Λ0)dΩlowast
Λ0 and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0) =
120583Λ0
Λ|xlowastΓΛ
(dΩlowastΛ0)dΩlowast
Λ0
The first key property of the measures 120583Λand 120583
Λ|xlowastΓΛ
isexpressed in the so-called FK-DLR equation We state it asLemma 21 below its proof repeats a standard argument usedin the classical case for establishing the DLR equation in afinite volume Λ sub Γ
Lemma 21 For all 119911 120573 gt 0 satisfying (22) and Λ
0sub Λ
1015840sub Λ
the probability density 119901Λ0
Λadmits the form
119901
Λ0
Λ(Ω
lowast
Λ0)
= int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840)120583
ΛΛ1015840
Λ(dΩlowast
ΛΛ1015840)
(100)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) = exp [minushΛ
0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(101)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast or
ΩlowastΛΛ1015840) and ΞΛ(Λ1015840
| ΩlowastΛΛ1015840) are determined as in (96)
Similarly for 119901Λ0
Λ|xlowastΓΛ
one has
119901
Λ0
Λ|xlowastΓΛ
(Ωlowast
Λ0) = int
119882lowast
ΛΛ1015840
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
times 120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840)
(102)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast
Λ0 | Ω
lowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 orΩ
lowast
ΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(103)
and the conditional partition functions ΞΛ(Λ
1015840 Λ
0| Ωlowast
Λ0 or
ΩlowastΛΛ1015840 or xlowast
ΓΛ) and Ξ
Λ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) are determined as in
(96)
As in [1] (100) and (102) mean that the conditional den-sities 119901Λ
0
Λ(Ωlowast
Λ0 | Ωlowast
ΛΛ1015840) and 119901
Λ0
Λ|xlowastΓΛ
(ΩlowastΛ0 | Ωlowast
ΛΛ1015840) relative
to 120590-algebra WΛΛ1015840
coincide respectively with 119902
Λ0
ΛΛ1015840(Ω
lowast0|
ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) for 120583ΛΛ
1015840
Λmdashand
120583ΛΛ1015840
Λ|xlowastΓΛ
mdashaaΩlowastΛΛ1015840 isin 119882
lowast
ΛΛ1015840 and aaΩlowast
Λ0 isin 119882
lowast
Λ0
As in [1] we call the expressions 119902Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) and
119902
Λ0
ΛΛ1015840(Ω
lowast
Λ0 | Ωlowast
ΛΛ1015840 or xlowast
ΓΛ) as well as the expressions
119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840) and 119902
Λ0
ΛΛ1015840(Ω
lowast0
| ΩlowastΛΛ1015840 or xlowast
ΓΛ)
appearing below the (conditional) RDM functionals (inbrief the RDMFs) The same name will be used for thequantity 119902Λ
0
ΛΛ1015840(Ω
lowast
Λ0 | Ω
lowast
ΛΛ1015840) from (110)-(111) and the quantity
119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) from (112)-(113)The second property is that the RDMKs FΛ
0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) are related to the measures 120583Λand 120583
Λ|xlowastΓΛ1015840
Again the proof of this fact is done by inspection
Lemma 22 The RDMK FΛ0
Λ(xlowast0 ylowast0) is expressed as follows
forallΛ
0sub Λ
1015840sub Λ
FΛ0
Λ(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
(104)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840)
(105)
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 15
Similarly
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 120572Λ(Ω
lowast0
)
times 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΛΛ1015840
120583ΛΛ1015840
Λ|xlowastΓΛ
(dΩlowastΛΛ1015840) 1 (Ωlowast
ΛΛ1015840 isin F
Λ0
)
times 119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(106)
where
119902
Λ0
ΛΛ1015840 (Ω
lowast0
| Ωlowast
ΛΛ1015840xlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ)
ΞΛ(Λ
1015840| Ωlowast
ΛΛ1015840 or xlowast
ΓΛ)
(107)
Here the partition functions ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
or ΩlowastΛΛ1015840) and
ΞΛ0
Λ(Λ
1015840 Λ
0| Ω
lowast0
orΩlowastΛΛ1015840 or xlowast
ΓΛ) are determined in (81) and
(83)
Remark 23 Summarizing the above observations the mea-sures 120583
Λand 120583
Λ|xlowastΓΛ
are concentrated on the subset in 119882
lowast
Λ
formed by loop configurations ΩlowastΛsuch that forall120591 isin [0 120573] the
sectionΩlowastΛ(120591) has le 120581 particles at each vertex 119894 isin Λ
3 The Class of Gibbs States G forthe Fock Space Model
31 Definition of Class G In this section we apply the ideafrom [1] to define the class of states G = G
119911120573for the model
introduced in Section 2 and state a number of results Theseresults will hold under condition (34) which is assumed fromnow on As in [1] the definition of a state 120593 isin G is based onthe notion of an FK-DLR probability measure 120583 on the space119882
lowast
Γ the class of these measures will be also denoted byG
Definition 24 Space 119882
lowast
Γis the (infinite) Cartesian product
times119894isinΓ
119882
lowast
119894(cf (60)) its elements are loop configurations Ωlowast
Γ=
Ωlowast(119894) 119894 isin Γ over Γ A component Ωlowast(119894) is a finite loopconfiguration (possibly empty) with an initialfinal particleconfiguration xlowast(119894) sub 119872 FormallyΩlowast(119894) is a finite collectionof loops Ω
lowast
119909119894 of time-length 120573119896
119909119894where 119896
119909119894= 1 2
starting and finishing at a point (119909 119894) isin 119872 times Γ For readerrsquosconvenience we repeat (37) for the case under consideration
Ω
lowast
119909119894 120591 isin [0 120573119896
119909119894] 997891997888rarr (119909 (Ω
lowast
119909119894 120591)
119894 (Ω
lowast
119909119894 120591)) isin 119872 times Γ
Ω
lowast
119909119894is cadlag Ω
lowast
119909119894(0) = Ω
lowast
119909119894(120573119896
119909119894minus) = (119909 119894)
Ω
lowast
119909119894has finitely many jumps on [0 120573119896
119909119894]
if a jump occurs at time 120591 then
d [119894 (Ωlowast
119909119894 120591minus )
119894 (Ω
lowast
119909119894 120591)] = 1
(108)
By W = WΓwe denote the 120590-algebra in 119882
lowast
Γgenerated
by cylindrical events Given a subset Γ sub Γ (finite or infinite)we denote byWΓ
= WΓ
Γthe 120590-subalgebra ofW generated by
cylindrical events localized in Γ Given a probability measure120583 = 120583
Γon (119882
lowast
ΓW
Γ) we denote by 120583Γ = 120583Γ
Γthe restriction of
120583 onWΓ
Definition 25 The class G under consideration is formed bymeasures 120583 which satisfy the following equation forall finite Λ sub
Γ and Λ
0sube Λ the probability density
119901
Λ0
(Ωlowast0) = 119901
Λ0
120583(Ω
lowast0) =
120583Λ0
Γ(dΩlowast0)
^ (dΩlowast0) Ω
lowast0isin 119882
lowast
Λ0
(109)
is of the form
119901
Λ0
(Ωlowast
Λ0) = int
119882lowast
ΓΛ
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)120583
ΓΛ(dΩlowast
ΓΛ) (110)
where
119902
Λ0
ΓΛ(Ω
lowast0| Ω
lowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΓ(Λ Λ
0| Ωlowast0 orΩlowast
ΓΛ)
ΞΓ(Λ | Ωlowast
ΓΛ)
(111)
and the conditional partition functions ΞΓ(Λ Λ
0| Ωlowast0 or
ΩlowastΓΛ
) and ΞΓ(Λ | Ωlowast
ΓΛ) are determined as in (96)
As in [1] (110) means that the conditional density119901
Λ0
|ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) relative to 120590-algebra WΓΛ coincides
with 119902
Λ0
ΓΛ(Ωlowast0 | Ωlowast
ΓΛ) for 120583ΓΛ
Γmdashaa Ωlowast
ΓΛisin 119882
lowast120573
ΓΛand ^
Λ0mdash
aaΩlowast0 isin 119882
lowast
Λ0
Remark 26 The measure 120583Γinherits the property from
Remark 23 and is concentrated on the subset in 119882
lowast
Γformed
by (infinite) loop configurations ΩlowastΓsuch that for all 120591 isin
[0 120573] the sectionΩlowastΛ(120591) hasle 120581 particles at each vertex 119894 isin Λ
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Advances in Mathematical Physics
Given ameasure 120583 isin G we associate with it a normalizedlinear functional 120593 = 120593
120583on the quasilocal 119862lowast-algebra B
First we set
FΛ0
(xlowast0 ylowast0)
= int
119882lowast
x0y0Plowast
x0 y0 (dΩlowast0
) 119861 (Ωlowast0
) 1 (Ωlowast0 isin FΛ0
)
times int
119882lowast
ΓΛ
120583ΓΛ
(dΩlowastΓΛ
) 1 (ΩlowastΓΛ
isin FΛ0
)
times 119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
(112)
where
119902
Λ0
ΓΛ(Ω
lowast0
| Ωlowast
ΓΛ)
= exp [minushΛ0
(Ωlowast0
| Ωlowast
ΓΛ)]
times
ΞΛ0
Λ(Λ Λ
0| Ω
lowast0
orΩlowastΓΛ
)
ΞΓ(Λ | Ωlowast
ΓΛ)
(113)
This defines a kernel FΛ0
(xlowast0 ylowast0) xlowast0 ylowast0 isin 119872
lowastΛ0
where Λ
0sub Γ is a finite set of sites It is worth reminding
the reader of the presence of the indicator functionals 1(sdot isinFΛ0
) in (112) and (113) (in the integral for ΞΛ0
Λ(Λ Λ
0|
Ωlowast0
or ΩlowastΓΛ
)) These indicators guarantee the compatibilityproperty forall finite Λ0
sub Λ
1
FΛ0
(xlowast0 ylowast0)
= int
119872lowastΛ1Λ0
prod
119895isinΛ1Λ0
prod
119911isinzlowast(119895)V (d119911)
times FΛ1
(xlowast0 or zlowastΛ1Λ0 ylowast0 or zlowast
Λ1Λ0)
(114)
Next we identify the operator RΛ0
(a candidate for theRDM in volumeΛ0) as an integral operator acting inH
Λ0 by
(RΛ0
120601) (xlowast0) = int
119872lowastΛ0
FΛ0
(xlowast0 ylowast0)120601 (ylowast0) dylowast0 (115)
Equation (114) implies that
trHΛ1Λ0
RΛ1
= RΛ0
(116)
Definition 27 The functional 120593 isin G is identified with the(compatible) family of operators RΛ
0
If the operators RΛ0
are positive definite (a property that is not claimed to beautomatically fulfilled) we again call it an FK-DLR state inthe infinite volume (for given values of activity 119911 and inversetemperature 120573) To stress the dependence on 119911 and 120573 wesometimes employ the notationG(119911 120573)
32 Theorems on Existence and Properties of FK-DLR StatesWe are now in position to state results about class G Weassume the conditions on the potentials 119880(1) and 119880
(2) fromthe previous section including the hard-core condition for119880
(1)
Theorem 28 For all 119911 120573 isin (0 +infin) satisfying (22) any lim-iting Gibbs state 120593 isin G0 (see Theorem 3) lies in G Thereforethe class of stateG s is nonempty
Theorem 29 Under condition (22) any FK-DLR state 120593 isin G
is G-invariant in the sense that forall finite Λ0sub Γ and forallg isin G
the RDM RΛ0
satisfies (35) Consequently (36) holds true
4 Proof of Theorems 3 6 28 and 29
41 Proof of Theorems 3 and 28 The proof is based on thesame approach as that used in [1] First given Λ
0sub Γ
we establish compactness of the sequence of the RDMKsFΛ0
Λ(xlowast0 ylowast0) and FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0) (see (78)ndash(83)) as functions
of variables xlowast0 = xlowast0(119894) ylowast0 = ylowast0(119894) isin 119872
lowastΛ0
with
♯xlowast0Λ
= ♯ylowast0Λ ♯xlowast0 (119894) ♯ylowast0 (119894) lt 120581 119894 isin Λ (117)
when Λ Γ Then we use Lemma 11 from [1] to derive thatthe sequence of the RDMs RΛ
0
Λand RΛ
0
Λ|xlowastΓΛ
is compact in thetrace-norm operator topology inH
Λ0
To verify compactness of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) we again as in [1] use the Ascoli-Arzela the-orem which requires the properties of uniform boundednessand equicontinuity These properties follow from the follow-ing
Lemma 30 (i) Under condition (22) the RDMKs FΛ0
Λ(xlowast0
ylowast0) and FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) admit the bounds
FΛ0
Λ(xlowast0 ylowast0) FΛ
0
Λ|xlowastΓΛ
(xlowast0 ylowast0)
le [(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(118)
where
Φ = sum
119896ge1
119911
119896 exp (119896Θ)
with Θ = 120581120573 (119880
(1)
+ 120581119880
(2)
+ 120581119869 (1) 119881)
(119)
(Note that Φ lt infin under the assumption (22)) Let 119901(119896)
119872yields
the supremum of the transition function over time 119896120573 for Brow-nian motion on the torus119872
119901
(119896)
119872= sup
119909119910isin119872
119901
119896120573(119909 119910) = 119901
119896120573(0 0) (120)
and 119901119872
= sup119896ge1
119901
(119896)
119872 Finally the upper-bound values 119880(1)
119880
(2) 119869(1) and 119881 have been determined in (14) (15) and (18)
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 17
(ii) The gradients of the RDMKs FΛ0
Λ(xlowast0 ylowast0) and
FΛ0
Λ|xlowastΓΛ
(xlowast0 ylowast0) satisfy forall119894 isin Λ and 119909 isin xlowast0(119894) 119910 isin ylowast0(119894)
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119909FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ(xlowast0 ylowast0)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
nabla119910FΛ0
Λ|xlowastΓΛ
(x0 y0)100381610038161003816100381610038161003816
le (♯Λ
0)ΘΦ
1015840119901119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(121)
where
Φ
1015840= sum
119896ge1
119896119911
119896 exp (119896Θ) (122)
(AgainΦ1015840lt infin under the condition (22)) The ingredients 119901
119872
and 119880
(1) 119880(2) 119869(1) and 119881 as in statement (i)
Proof of Lemma 30 First observe that119901119872
lt infinon a compactmanifold Bound (119) is established in a direct fashion Firstwe majorize the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0) = hΛ
0
(Ωlowast0
) + h (ΩlowastΛΛ0 || Ω
lowast0
) (123)
contributing to the RHS in (80) and (81) and the energy
hΛ0
(Ωlowast0
| Ωlowast
ΛΛ0 or xlowast
ΓΛ)
= hΛ0
(Ωlowast0
| xlowastΓΛ
) + h (ΩlowastΛΛ0 || Ω
lowast0
)
(124)
contributing to the RHS in (82) and (83)This yields the factor
prod
120596isinΩlowast0
119911
119896(120596lowast
) exp [119896 (120596lowast)Θ] (125)
Next we majorize the integral = int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0(dΩlowast0
) in(80) and (82) this gives the factor
(♯Λ
0) 119901
119872[(120581♯Λ
0)] (119901
119872)
120581♯Λ0
Φ
♯Λ0
(126)
The aftermath are the ratios (78) and (79) with xlowast0 = ylowast0 = 0they do not exceed 1
Passing to (121) let us discuss the gradients nabla119909only (The
gradients in the entries of nabla119910are included by symmetry)
The gradient in (121) of course affects only the numeratorsΞΛ0
Λ(xlowast0 ylowast0 Λ Λ
0) and Ξ
Λ0
Λ(xlowast0 ylowast0 Λ Λ
0| xlowast
ΓΛ) in (78)
and (79) The bounds (121) are done essentially as in [1] Fordefiniteness we discuss the case of the RDMK FΛ
0
Λ(xlowast0 ylowast0)
the RDMK FΛ0
Λ|xlowastΓΛ
(x0 y0) is treated similarly There are twocontributions into the gradient one comes from varyingthe measure Plowast
xlowast0ylowast0(dΩlowast0
) and the other from varying thefunctional exp[minushΛ
0
(Ωlowast0
) minus hΛΛ0
(ΩlowastΛΛ0 | Ω
lowast0
)]The first contribution can again be uniformly bounded in
terms of the constant 119901119872 The detailed argument as in [1]
includes a deformation of a trajectory and is done similarly
to [1] (the presence of jumps does not change the argumentbecause 119901
119872yields a uniform bound in (43))
The second contribution yields again as in [1] an expres-sion of the form
int
119882lowast
xlowast0ylowast0Plowast
xlowast0 ylowast0 (dΩlowast0
)
times sum
119894isinΛ0
sum
119909isinxlowast0(119894)
h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
Ωlowast
ΛΛ0)
times exp [minushΛ0
(Ωlowast0
) minus hΛΛ0
(Ωlowast
ΛΛ0 | Ω
lowast0
)]
(127)
where the functional h119909119894(Ω
lowast
(119909119894)1205740(119909119894)
ΩlowastΛΛ0) is uniformly
bounded Combining this an upper bound similar to (119)yields the desired estimate for the gradients in (121)
Hence we can guarantee that the RDMs RΛ0
Λand RΛ
0
Λ|xlowastΓΛ
converge to a limiting RDM RΛ0
along a subsequence in Λ
Γ The diagonal process yields convergence for every finiteΛ
0sub Γ A parallel argument leads to compactness of the
measures 120583Λ0
Λfor any given Λ
0 as Λ Γ We only give herea sketch of the corresponding argument stressing differenceswith its counterpart in [1]
In the probabilistic terminology measures 120583Λrepresent
random marked point fields on 119872 times Γ with marks from thespace119882lowast
= 119882
lowast
0where119882lowast
0= ⋃
119896ge1119882
119896120573
0and119882119896120573
0is the space
of loops of time-length 119896120573 starting and finishing at 0 isin 119872 andexhibiting jumps that is changes of the index (The space119882lowast
119909119894
introduced inDefinitions 7 and 13 can be considered as a copyof 119882lowast placed at site 119894 isin Γ and point 119909 isin 119872) The measure120583
Λ0
Λdescribes the restriction of 120583
Λto volume Λ0 (ie to the
sigma algebra Wlowast
Λ0) and is given by its Radon-Nikodym
derivative 119901
Λ0
Λrelative to the reference measure dΩlowast
Λ0 on
119882
lowast
Λ0 (cf (66) (100)) The reference measure is sigma-finite
Moreover under the condition (22) the value 119901
Λ0
Λ(Ωlowast
Λ0) is
uniformly bounded (in both Λ Γ and ΩlowastΛ0 isin 119882
lowast
Λ0)
This enables us to verify tightness of the family of measures120583
Λ0
Λ Λ Γ and apply the Prokhorov theorem Next we
use the compatibility property of the limit-point measures120583
Λ0
Γand apply the Kolmogorov theorem This establishes the
existence of the limit-point measure 120583Γ
By construction and owing to Lemmas 21 and 22 thelimiting family RΛ
0
yields a state belonging to the class GThis completes the proof of Theorems 3 and 28
42 Proof ofTheorems 6 and 29 Theassertion ofTheorem 6 isincluded inTheorem 29Therefore wewill focus on the proofof the latterThe proof based on the analysis of the conditionalRDMFs 119902Λ
0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) and 119902
Λ0
ΓΛ(Ω
lowast0
| ΩlowastΓΛ
) introducedin (111) and (112) For definiteness we assume that vertex 119900 isin
Λ
0 so that Λ0 lies in the ball Λ119899for 119899 large enough As in
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Advances in Mathematical Physics
[1] the problem is reduced to checking that forall119911 120573 isin (0infin)
satisfying (22) g isin G and finite Λ0sub Γ
lim119899rarrinfin
119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
)
119902
Λ0
ΓΛ(119899)(Ω
lowast0
| ΩlowastΓΛ(119899)
)
= 1 (128)
here we need to establish this convergence (128) uniformly inthe argumentΩlowast
ΓΛ(119899)= Ωlowast(119894) 119894 isin Γ Λ(119899)with ♯Ωlowast(119894) le 120581
and in Ωlowast0 outside a set of the Plowast120573
x0 y0 measure tending to 0
as 119899 rarr infin The latter is formed by path configurations Ωlowast0
that contain trajectories visiting sites 119894 isin Γ Λ(119903(119899)) where119903(119899) grows with 119899 see Lemma 31 belowThe action of g upona path configuration Ωlowast0 = Ω
lowast
(119909119894)1205740(119909119894)
119894 isin Λ
0 119909 isin xlowast0 is
defined by
gΩlowast0
= gΩlowast
(119909119894)1205740(119909119894)
where (gΩlowast
(119909119894)1205740(119909119894)
) (120591) = g (Ωlowast
(119909119894)1205740(119909119894)
(120591))
(129)
We want to establish that forall119886 isin (1infin) for any 119899 largeenough the conditional RDMFs satisfy
119886119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| Ωlowast
ΓΛ(119899)) + 119886119902
Λ0
ΓΛ(119899)(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))
ge 2119902
Λ0
ΓΛ(119899)(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(130)
As in [1] we deduce (130) with the help of a special con-struction of ldquotunedrdquo actions g
Λ(119899)Λ0 on loop configurations
120596Λ(119899)Λ
0 (over which there is integration performed in thenumerators
ΞΛ0
Γ(Λ (119899) Λ
0| gΩ
lowast0
orΩlowast
ΓΛ(119899))
ΞΛ0
Γ(Λ (119899) Λ
0| g
minus1Ω
lowast0
orΩlowast
ΓΛ(119899))
(131)
in the expression for 119902
Λ0
ΓΛ(119899)(gΩ
lowast0
| ΩlowastΓΛ(119899)
) and119902
Λ0
ΓΛ(119899)(gminus1Ω
lowast0
| ΩlowastΓΛ(119899)
)) The tuning in gΛ(119899)Λ
0 is chosenso that it approaches e (or the 119889-dimensional zero vector inthe additive form of writing) the neutral element of G whilewe move from Λ
0 towards Γ Λ(119899)Formally (130) follows from the estimate (132) below forall
finite Λ
0sub Γ Ωlowast0 isin 119882
Λ(119899) g isin G and 119886 isin (1infin) for
any 119899 large enough forallΩlowastΛ(119899)Λ
0 = Ωlowast(119894) 119894 isin Λ(119899) Λ
0 and
ΩlowastΓΛ(119899)
= Ωlowast(119894) 119894 isin Γ Λ(119899) with ♯Ωlowast(119894) le 120581
119886
2
exp [minushΛ(119899) ((gΩlowast0) or (gΛ(119899)Λ
0Ωlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [ minus hΛ(119899)
times ((gminus1Ω
lowast0
) or (gminus1
Λ(119899)Λ0Ω
lowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(132)
In (132) the loop configuration gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 is deter-mined by specifying its temporal section g
Λ(119899)Λ0Ω
lowast
119909119894(120591 +
120573119898) 119894 isin Λ(119899) Λ
0 119909 isin xlowast(119894) 0 le 119898 lt 119896
119909119894 That is we
need to specify the sections
(119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) 119897 (120591 + 120573119898 g
Λ(119899)Λ0Ω
lowast
119909119894)) (133)
for loopsΩlowast
119909119894constituting g
Λ(119899)Λ0Ωlowast
Λ(119899)Λ0 To this end we set
119906 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = g
(119899)
119895119906 (120591 + 120573119898Ω
lowast
119909119894)
if 119897 (120591 + 120573119898 gΛ(119899)Λ
0Ω
lowast
119909119894) = 119895
(134)
In other words we apply the action g(119899)119895
to the temporal sec-tions of all loops Ω
lowast
119909119894located at vertex 119895 at a given time
regardless of position of their initial points (119909 119894) inΛ(119899) Λ
0Observe that (130) is deduced from (132) by integrating
in dΩlowastΛ(119899)Λ
0 and normalizing by ΞΛ(119899)Λ
0(ΩlowastΓΛ(119899)
) see (80)with Λ
1015840= Λ(119899) (The Jacobian of the map Ωlowast
Λ(119899)Λ0 997891rarr
gΛ(119899)Λ
0ΩlowastΛ(119899)Λ
0 equals 1)Thus our aim becomes proving (132) The tuned family
gΛ(119899)Λ
0 is composed of individual actions g(119899)119895
isin G
gΛ(119899)Λ
0 = g(119899)
119895 119895 isin Λ (119899) Λ
0 (135)
Elements g(119899)119895
are powers (multiples in the additive parlance)of element g isin G figuring in (128)ndash(132) (resp of the corre-sponding vector 120579 isin 119872 cf (9)) and defined as follows Let120579
(119899)
119895denote the vector from119872 corresponding to g(119899)
119895 and we
select positive integer values 119903(119899) = lceillog (1 + 119899)rceil and set
120579
(119899)
119895= 120579120592 (119899 119895) (136)
where
120592 (119899 119895) =
1 d (119900 119895) le 119903 (119899)
120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899)) d (119900 119895) gt 119903 (119899)
(137)
In turn the function 120599 is chosen to satisfy
120599 (119886 119887) = 1 (119886 le 0) +
1 (0 lt 119886 lt 119887)
119876 (119887)
times int
119887
119886
119911 (119906) d119906 119886 119887 isin R
(138)
with
119876 (119887) = int
119887
0
120577 (119906) d119906 sim log log 119887
where 120577 (119906) = 1 (119906 le 2) + 1 (119906 gt 2)
1
119906 ln 119906 119887 gt 0
(139)
Lemma 31 Given 119911 120573 isin (0infin) satisfying (22) and a finiteset Λ0 there exists a constant 119862 isin (0infin) such that forallx0 y0 isin
119872
lowastΛ0
the set of path configurations Ωlowast0 isin 119882
lowast
xlowast0 ylowast0 withhΛ0
(Ωlowast0
) lt +infin which include trajectories visiting points inΓΛ(119903(119899)) has thePlowast
x0 y0 measure that does not exceed119862(lceil(1+119903(119899))rceil)
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Mathematical Physics 19
Proof of Lemma 31 The condition that hΛ0
(Ωlowast0
) lt +infin
implies that the total number of sub trajectories of length120573 in Ωlowast0 does not exceed 120581 times ♯Λ
0 which is a fixed valuein the context of the lemma Each such trajectory has aPoisson number of jumps this produces the factor 1(lceil(1 +
119903(119899))rceil)
Back to the proof of Theorem 29 let gminus1Λ(119899)Λ
0 be the col-lection of the inverse elements
gminus1
Λ(119899)Λ0 = g
(119899)
119895
minus1
119895 isin Λ (119899) Λ
0 (140)
The vectors corresponding to g(119899)119895
minus1
are minus120579(119899)119895
isin 119872 We will
use this specification for g(119899)119895
and g(119899)119895
minus1
for 119895 isin Λ(119899) or evenfor 119895 isin Γ as it agrees with the requirement that g(119899)
119895equiv gwhen
119895 isin Λ
0 and g(119899)119895
equiv e for 119895 isin Γ Λ(119899) Accordingly we will usethe notation g
Λ(119899)= g(119899)
119895 119895 isin Λ(119899)
Observe that the tuned family gΛ(119899)Λ
0 does not changethe contribution into the energy functional hΛ(119899)|ΓΛ(119899) com-ing from potentials119880(1) and119880(2) it affects only contributionsfrom potential 119881
The Taylor formula for function 119881 together with theabove identification of vectors 120579(119899)
119895 gives
100381610038161003816100381610038161003816
119881 (g(119899)
119895119909 g
(119899)
1198951015840119909
1015840) + 119881(g
(119899)
119895
minus1
119909 g(119899)
1198951015840
minus1
119909
1015840) minus 2119881 (119909 119909
1015840)
100381610038161003816100381610038161003816
le 119862
1003816100381610038161003816
120579
1003816100381610038161003816
210038161003816100381610038161003816
120592 (119899 119895) minus 120592 (119899 119895
1015840)
10038161003816100381610038161003816
2
119881 119909 119909
1015840isin 119872
(141)
Here 119862 isin (0infin) is a constant |120579| stands for the norm of thevector 120579 representing the element g and the value 119881 is takenfrom (14)
Next the square |120592(119899 119895) minus 120592(119899 119895
1015840)|
2 can be specified as
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
=
0
if d (119895 119900) d (1198951015840 119900) le 119903 (119899)
0
if d (119895 119900) d (1198951015840 119900) ge 119899
[120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))]
2
if 119903 (119899) lt d (119895 119900) d (1198951015840 119900) le 119899
120599(d(119895 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (119895 119900) le 119899 d (1198951015840 119900) isin ]119903 (119899) 119899[
120599(d(1198951015840 119900) minus 119903(119899) 119899 minus 119903(119899))
2
if 119903 (119899) lt d (1198951015840 119900) le 119899 d (119895 119900) isin ]119903 (119899) 119899[
(142)
By using convexity of the function exp and (141) forall119886 gt 1
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minus12
hΛ(119899)|ΓΛ(119899)
times (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) Ω
lowast
ΓΛ(119899))
minus
1
2
hΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge 119886 exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))] 119890
minus119862Υ2
(143)
where
Υ = Υ (119899 g) = 120573120581
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840))
10038161003816100381610038161003816
120592(119899 119895) minus 120592(119899 119895
1015840)
10038161003816100381610038161003816
2
(144)
The next observation is that
Υ le 3120573120581
21003816100381610038161003816
120579
1003816100381610038161003816
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
1 (d (119895 119900) le d (1198951015840 0)) 119869 (d (119895 119895
1015840))
times [120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))]
2
(145)
where owing to the triangle inequality for all 119895 1198951015840 d(119895 119900) led(1198951015840 119900)
0 le 120599 (d (119895 119900) minus 119903 (119899) 119899 minus 119903 (119899))
minus 120599 (d (1198951015840 119900) minus 119903 (119899) 119899 minus 119903 (119899))
le d (119895 1198951015840)
120577 (d (119895 119900) minus 119903)
119876 (119899 minus 119903 (119899))
(146)
This yields
Υ le 120573120581
23
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
times sum
(1198951198951015840
)isinΛ(119899)timesΓ
119869 (d (119895 1198951015840)) d(119895 119895
1015840)
2
120577(d(119895 0) minus 119903 (119899))
2
le
3
1003816100381610038161003816
120579
1003816100381610038161003816
2
119876(119899 minus 119903(119899))
2
[
[
sup119895isinΓ
sum
1198951015840isinΓ
1198691198951198951015840d(119895 119895
1015840)
2]
]
times sum
119895isinΛ119899+1199030
120577(d (119895 0) minus 119903 (119899))
2
(147)
where function 120577 is determined in (139)
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 Advances in Mathematical Physics
Therefore it remains to estimate the sumsum
119895isinΛ119899+1199030
120577(d(119895 0) minus 119903(119899))
2 To this end observe that 119906120577(119906) lt 1
when 119906 isin (3infin) The next remark is that the number of sitesin the sphere Σ
119899grows linearly with 119899 Consequently
sum
119895isinΛ119899+1199030
120577(d(119895 119900) minus 119903(119899))
2
= sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
times sum
119895isinΣ119896
120577 (119896 minus 119903 (119899))
le 1198620
sum
1le119896le119899+1199030
120577 (119896 minus 119903 (119899))
le 1198621119876 (119899 + 119903
0minus 119903 (119899))
Υ le
119862
119876 (119899 minus 119903 (119899))
997888rarr infin as 119899 997888rarr infin
(148)
Therefore given 119886 gt 1 for 119899 large enough the term119886119890
minus119862Υ2 in the RHS of (145) becomes gt1 Hence
119886
2
exp [minushΛ(119899) (gΛ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
+
119886
2
exp [minushΛ(119899) (gminus1Λ(119899)
(Ωlowast0
orΩlowast
Λ(119899)Λ0) | Ω
lowast
ΓΛ(119899))]
ge exp [minushΛ(119899) (Ωlowast0 orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ(119899))]
(149)
Equation (149) implies that the quantity
119902
Λ0
|ΓΛ(119899)(Ω
lowast0
| ΩΓΛ(119899)
)
= int
119882Λ(119899)Λ0
dΩlowastΛ(119899)Λ
0
times
exp [minushΛ0
(Ωlowast0
orΩlowastΛ(119899)Λ
0 | Ωlowast
ΓΛ0)]
ΞΛ(119899)
(ΩlowastΓΛ(119899)
)
(150)
obeys
lim119899rarrinfin
[119902
Λ0
|ΓΛ(119899)
120573(gΩ
lowast0
| Ωlowast
ΓΛ(119899))
+119902
Λ0
|ΓΛ(119899)
120573(g
minus1Ω
lowast0
| Ωlowast
ΓΛ(119899))]
ge 2 lim119899rarrinfin
119902
Λ0
|ΓΛ(119899)
120573(Ω
lowast0
| Ωlowast
ΓΛ(119899))
(151)
uniformly in boundary condition 120596ΓΛ(119899)
Integrating (151)d120583ΓΛ(119899)
Γ(120596
ΓΛ(119899)) yields (132)
Acknowledgments
This work has been conducted under Grant 201120133-0provided by the FAPESP Grant 20115764450 provided byThe Reitoria of the Universidade de Sao Paulo and Grant201204372-7 provided by the FAPESP The authors expresstheir gratitude to NUMEC and IME Universidade de SaoPaulo Brazil for the warm hospitality
References
[1] M Kelbert and Y Suhov ldquoA quantumMermin-Wagner theoremfor quantum rotators on two-dimensional graphsrdquo Journal ofMathematical Physics vol 54 no 3 Article ID 033301 2013
[2] Y Kondratiev Y Kozitsky and T Pasurek ldquoGibbs random fieldswith unbounded spins on unbounded degree graphsrdquo Journal ofApplied Probability vol 47 no 3 pp 856ndash875 2010
[3] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem for Gibbs states on Lorentzian triangulationsrdquo Journalof Statistical Physics vol 150 pp 671ndash677 2013
[4] M Kelbert Yu Suhov and A Yambartsev ldquoA Mermin-Wagnertheorem on Lorentzian triangulations with quantum spinsrdquoJournal of Statistical Physics vol 150 no 4 pp 671ndash677 2013
[5] H Tasaki ldquoThe Hubbard modelmdashan introduction and selectedrigorous resultsrdquo Journal of Physics vol 10 no 20 p 4353 1998
[6] J Hubbard ldquoElectron correlations in narrow energy bandsrdquoProceedings of the Royal Society A vol 276 pp 238ndash257 1963
[7] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013
[8] Y Suhov and M Kelbert ldquoFK-DLR states of a quantum bose-gasrdquo httparxivorgabs13040782
[9] Y Suhov M Kelbert and I Stuhl ldquoShift-invariance for FK-DLR states of a 2D quantum bose-gasrdquo httparxivorgabs13044177
[10] R A Minlos A Verbeure and V A Zagrebnov ldquoA quantumcrystal model in the light-mass limit gibbs statesrdquo Reviews inMathematical Physics vol 12 no 7 pp 981ndash1032 2000
[11] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol I Functional Analysis Academic Press New YorkNY USA 1977
[12] O Bratteli and D Robinson Operator Algebras and Quantum-Statistical Mechanics Vol I Clowast- and Wlowast-Algebras SymmetryGroups Decomposition of States Springer Berlin Germany2002
[13] O Bratteli and D Robinson Operator Algebras and QuantumStatistical Mechanics Vol II Equilibrium States Models inQuantum Statistical Mechanics Springer Berlin Germany2002
[14] K I Sato Levy Processes and Infinitely Divisible DistributionsCambridge University Press New York NY USA 2011
[15] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol II Fourier Analysis Academic Press NewYorkNYUSA 1972
[16] M Reed and B Saımon Methods of Modern MathematicalPhysics Vol IV Analysis of Operators Academic Press NewYork NY USA 1977
[17] B Simon Functional Integration and Quantum Physics Aca-demic Press New York NY USA 1979
[18] J Ginibre ldquoSome applications of functional integration inStatistical mechanicsrdquo in Statistical Mechanics and QuantumField Theory C M de Witt and R Stora Eds pp 327ndash428Gordon and Breach New York NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of