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Large deviations of SLE and Weil PeterssonTeichmillerspaceLecture 3 Yilin Wang MT
Loewher energy a
SLE A Weil PeterssonTeichmiller spanSchramin Leewner
tevolutions
ADeterministicProbabilistic
Stochastic analysis quasicircles
Conformally invariant quasiconformal mapping2D Statistical mechanics
Geometricfunction theorymodelComplex structures
Random planar maps on Riemannsurfaces
2D quantum gravity Kahler geometryConformal field theory String theory
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Monday
1 Brownian Motion and Dirichlet energySchilder's theorem
I SLE and Loewner energyF RB Icr I wait dt
4 SLE large deviations
Energy reversibility from SLE reversibility
Tuesday
2 Loop energyGeneralizes chordal energy
I Weil Petersson Teichmiller space248 a J 8 is Weil Peterssonand 25 other equivalent definitions
Today1 Radial SLE a large deviations
1 Foliations by Weil Petersson quasicircles
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Motivation SLE duality
For K 78Dubédat Ethan Miller Sheffield
ylocally a SLE a
www.odftfo gEk
Asymptoticbehavior of Slee
dualShea
WPquasicircle Leewner energy c
V W LaettnerKufuor energy
Q What happens when we let k o
For SLEK.KZ EH2
9114 Gees fBy gotZ
Ect maximal solution time
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I toys
td.ttHt 9 Z E1H TA t 3
domainof definitionof go 1418 to t
Can show Gt itk
y z
Not so interesting
Issue normalized at a boundarypoint too
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1 Radial SLE Large deviations
1 Loewner Kufra equationNy Pelt o Pt EProbis
measurable in t
Loewner Kufarev equation ZED
4ft'D Ztily two II jPDE
folz Z TIE 0
ft ID Dt with ft lo o ft cos et
A not ft too Evolution family Dt
The normalvelocity of 2Dt at file is
27Pelo file if Prado Pt
Stift satisfies
CODE at9 12 g iz fe 9 14e geez
Pt'do
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Examples
Ptldo I do f t yo
yDtt
of
De é'D
f Dirac measure
ft Ido I Sei Bt Brownian motion on R
MI ack28Radial
ni i t.ggSEKp Dt Dt
so
2 K soo limit
Let K 70 in ft Ido e Sei But
Is trot
09th Jett gigs eh Seibu idol ds
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I 9 12 gots gaz
Littorio IO do
occupation
T t go.izfs.eeog I y measure up
to time t
at Getz
2 9 12 9 12 Gold Z
gait et z De ID
DtD
Illustration of occupation measure
to
tangoEmoteuponto
S x iRt 3h10 t
Seibaldodt dy It
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3 Radial SLE LDP
Thm Ang Park W Io
Radial stem process satisfies the LDP as k so
with rate function St CLeewner Kufarer energy
IP I SLEK Dit o expL KSelps as k a
whereSeip J Lepe dt
Lepe Ifs Ive cost do
if Pt Dt lo do and left o otherwise
Followsfrom Donsker Vardhan theorem on
the LDP of occupation measures
Stlp so only for a c measures
on S x RtMore regular than Dirac masses
Q What are the families Dt generated byPt where Step co
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I Foliations by Weil Petersson quasicircles
Whole plane Loewnerkafana equation
Pt Pt lean Dt teaand ft D De withft't o ftlie e
t
such that
chain generated
is the Leever tape
0 by Pas thoID
Sip r f Lepe de
Claim LoewnerKufuor chain in ID is
a special case of whole plan L K chain
Given Pt ex
Set Pt do for all to
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Do ID
fDe et'D for tooDotty is the family generated
by Pt too
Then Vikland W
If Sep co then
2Dt is aWeil Petersson quasicircle ft er
U JDt 61103t t 2Dt is continuous in the sup norm
Foliation of Ello by Weil Petersson quasicircles
a Non smoothMonotone but not strictly monotone2Dt is called a leaf
Finite L K energy foliation has finite Leewner
energy leaves
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Wewill prove it by showing aquantitative result
Recall g t ft De D
Define Y iz arg giltst
go12
if Z t ODt
z
Y is the winding function ofthe foliation generated by Pt tar
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Than V W
ab Stp Jeter day Die
ab is consistent with SLE duality
K as In
Example
Pt do I sin E do for te to
t I do otherwise
foliation 4Rt
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Corollary Sips es Ein
xD
ID feces LP
Pr generates the foliation formedby equipotentialab Sip7 142 21g I
Y is harmonic in 618
We recall I 8 If pargf dat fye Tarp Fda
41 14 1 I
fkn o
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Question from T Amaba
16 from SLE duality
IP SLEa loop stays close to 8
a
exp l ÉSI
ko
Pld SLE It stays close too
SI
exp I k If sips11 IDE 8
expl I Sip's
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Cor W Definition 27AJordan curve 8 separating o and a is Weil Petersson
8 can be realized as a leaf inthe foliation generated by a measure
with Stp r
Proof follows from previouscorollary
248 set Sept em Sip
y
inA
winding function
yry a y
harmonic in 618since leg 51 no true for all Jordan
curvesseparating
o and a
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Reversibility of Loaner kafana energy
p y5 I
Sip SipProof Yes yes DII
Remark
Reversibility of radial slew forK 8 is not known
This resultsuggests
it to be true
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Proof sketch of 4 1681ps Dce
Dirichlet energy is conformally invariant
jLoenner chain Explore a conformallyinvariant object layerbylayer
Assume Pete generates a foliation of ID
of a function on ID sit D O a
op o few0 any
ahit
e poet4 Aw Dtharmonicin De
o outsideofDt
p Pt idol dt
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Thin VW Disintegration isometry
WED D SxiR 28
u Ian tfis an bijective isometry with inverse operator
in
harmonic function in Dt
Aconsequence GEF Whitenoise decamp generalizes
IHedenmalm Nieminen
Proof of 4
If 9 a winding function
p Eco do dt2WEshow v4 ont ut
DIY44 Yup Jfs 4kt4 2 24 IO do It
161 Lapeldt lbStp
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Conclusion
StenKoot
Largedeviations
Loewner energyI'in
t
SLE
duality conformal frappeLDP Diy duality byWeil
geometryK or
Petersson
sips
quasicircles
largeammoLeewner
knfarer energySLEWox
Wang 2021