statistics of real eigenvalues in ginoe spectra snowbird conference on random matrix theory &...

Statistics of Real Eigenvalues in GinOE Spectra

Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007

Eugene KanzieperDepartment of Applied

MathematicsH.I.T. - Holon Institute of

TechnologyHolon 58102, Israel

Gernot Akemann (Brunel)

Phys. Rev. Lett. 95, 230501 (2005)arXiv: math-ph/0703019 (J. Stat. Phys.)

Applied Mathematics

Statistics of Real Eigenvaluesin GinOE Spectra

42

in preparation

Alexei Borodin(Caltech)

[ ]

20052007

What is the probability that an n × n random real matrix with Gaussian i.i.d. entries has

exactly k real eigenvalues?A. Edelman (mid-nineties)

Statistics of Complex Spectra

Applied Mathematics41

Statistics of Real Eigenvaluesin GinOE Spectra [ ]

» The Problem

Ginibre’s random matrices • Definitions & physics applications

Ginibre’s real random matrices (GinOE)

• Overview of major developments since 1965

• Real vs complex eigenvalues: What is (un)known ?

Conclusions & What is next ?

• Probability to find exactly k real eigenvalues and inapplicability of the Dyson integration theorem



» Outline

• Pfaffian integration theorem

GUEHGinUE

GSEHGinSE

GOEHGinOE

)2( :

)4( :

)1( :

NN

NN

NN

C

Q

R

1965

1 1,

)(2/)( tr exp)(2

q

N

ji

qij

NN dP HHHHH D

com

ple

xit

y

s u

c c

e s

s

Statistics of Complex SpectraIs there any

physics

Dropped Hermiticity…

?



» Ginibre’s random matrices: also physics

?Is there any

physics




• Dissipative quantum chaos (Grobe and Haake 1989) • Dynamics of neural networks (Sompolinsky et al 1988, Timme et al 2002, 2004)• Disordered systems with a direction (Efetov 1997)• QCD at a nonzero chemical potential (Stephanov 1996)• Integrable structure of conformal maps (Mineev-Weinstein et al 2000)• Interface dynamics at classical and quantum scales (Agam et al 2002)• Time series analysis of the brain auditory response (Kwapien et al 2000)• More to come: Financial correlations in stock markets (Kwapien et al 2006)




?Is there any

physics

<< 1

GinOE model

ASL HghHH

~ 1

directed chaos




Applied Mathematics

Universal noise dressing

is still there !

Asymmetric L-RCross-Correlation

Matrices

35Statistics of Real Eigenvalues

in GinOE Spectra [ ]» Ginibre’s random matrices: also physics

Applied Mathematics

• Dissipative quantum chaos (Grobe and Haake 1989) • Dynamics of neural networks (Sompolinsky et al 1988, Timme et al 2002, 2004)• Disordered systems with a direction (Efetov 1997)• QCD at a nonzero chemical potential (Stephanov 1996)• Integrable structure of conformal maps (Mineev-Weinstein et al 2000)• Interface dynamics at classical and quantum scales (Agam et al 2002)• Time series analysis of the brain auditory response (Kwapien et al 2000)• More to come: Financial correlations in stock markets (Kwapien et al 2006)

Back to 1965 and Ginibre’s maths curiosity…


in GinOE Spectra [ ]» Ginibre’s random matrices: also physics



» Outline reminder








1965

1 1,

)(2/)( tr exp)(2

q

N

ji

qij

NN dP HHHHH D

GinUEGinSE

(almost) uniformdistribution

depletion fromreal axis

accumulation along real axis



» Spectra of Ginibre’s random matrices

GinOE

1965

GinUEGinSE

GinOE







1965

GinUE


N

k

zzN

kkkkNN

kkezzNCzzP1

2

21

21

21)(,...,

GinUE: jpdf + correlations




1965


N

k

zzkk

N

kkkkkkNN

kkezzzzzzNCzzP1

222

41

21

2121)(,...,

GinSE


GinSE: jpdf+ correlations

Mehta, Srivastava 1966




1965


GinSE: jpdf+ correlations

Mehta, Srivastava 1966

GinOE


N

j

N

jjjjNNNTH

jewwP1

2/ 1 )/(

2

21

21 ),,(




1965

GinOE


N

j

N

jjjjNNNTH

jewwP1

2/ 1 )/(

2

21

21 ),,(

…

Key Feature )(NTH

)(NT

)/( ... )/( ... )0/( NNTkNTNTN

k

kNTNT1

)/()(

0 number of real

eigenvalues

N

kNkNTHNNTH wwPwwP

1 1 )/( 1 )( ),,(),,(

0 ?






» Outline reminder








})({)/( wP NNTH

196

5

Ginibre

1991

Lehmann & Sommers

2/1

1 ,)/(

2erfc })({

2

jjN

j

wN

jijikNkNTH

wwewwCwP j

1997

Edelman

1994

Edelman, Kostlan & Shub

2

1 1

2][ E 1

,0

nkN

N

k

On

pkk

quarter of a century !!

CorrelationFunctions ?!

}{ seigenvalue c.c. of pairs

1 1

seigenvalue real

1 1 ,,,,,,, ,, l

ll zzzzwwwk

kN

N

kNkNTHNNTH wwPwwP

1 1 )/( 1 )( ),,(),,(

0



» Overview of major developments since 1965



» Overview of major developments since 1965

196

5

Ginibre

1991

Lehmann & Sommers

1997

Edelman

1994

Edelman, Kostlan & Shub

quarter of a century !!

CorrelationFunctions ?!

N

kNkNTHNNTH wwPwwP

1 1 )/( 1 )( ),,(),,(

0

Borodin & Sinclair, arXiv: 0706.2670

Forrester & Nagao, arXiv: 0706.2020

Sommers, arXiv: 0706.1671detailed k-th partial correlation functions are not available…



» Outline reminder








1997

Edelman

Probability to have all eigenvalues real

4/)1(, 2 NNNNp (the smallest one)

Theorem

2,,, kNkNkN srp ( rational)kNkN sr ,, &

})({ )/(1

, wPdwp kNTH

N

jjkN



» Real vs complex eigenvalues

1997

Edelman

+l2kN 4/)1(

, 2 NNNNp 2,,, kNkNkN srp

k

j ppjpj

qpqpqp

k

jiji

zzpp

Z p

ppp

k

jj

kNkN

zzzzzz

ei

zz

i

zzZded

k

cp ppj

1 1

2 2

2/)(

0}Im{ 1

22/

1

,,

2

erfc 2

!!

2 2 2

ll

l

l

Solved ?.. }{

seigenvalue c.c. of pairs

1 1

seigenvalue real

1 1 ,,,,,,, ,, l

ll zzzzwwwk

kN

})({ )/(1

, wPdwp kNTH

N

jjkN




4/)1(, 2 NNNNp

MATHEMATICA code up to

9N

No ClosedFormula for

kNp ,






» Outline reminder








ˆ 1̂ det )( ,2,

]2/[

0

zppzzG NNNN

N

N ll

l Even Better

),,( 1,, llF xxpp NNkN

Starting point

k

j ppjpj

qpqpqp

k

jiji

zzpp

Z p

ppp

k

jj

kNkN

zzzzzz

ei

zz

i

zzZded

k

cp ppj

1 1

2 2

2/)(

0}Im{ 1

22/

1

,,

2

erfc 2

!!

2 2 2

ll

l

l

The Answer a probability to have all eigenvalues real

universal multivariate polynomials

)(

}{ 11 !

1 )1(),,(

l

l

llll l

Fp g

j jj

j

jx

xx

gg 2 1 ,, , 21

llll

integer partitions

jNjx

)1]2/[ ,0( ˆ tr

a nonuniversal ingredient

zonal polynomialsJack polynomials at α=2

l2kN

),...,(!

1),...,( 1)1(1 lll lZ

lF xxxx



» Probability to find exactly k real eigenvalues

No visible discrepancies with numeric simulations over 10 orders of magnitude !!



» Probability to find exactly k real eigenvalues

0 2 4 6 8 10 121. 1010

1. 108

1. 106

0.0001

0.01

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

k

j ppjpj

qpqpqp

k

jiji

zzpp

Z p

ppp

k

jj

kNkN

zzzzzz

ei

zz

i

zzZded

k

cp ppj

1 1

2 2

2/)(

0}Im{ 1

22/

1

,,

2

erfc 2

!!

2 2 2

ll

l

l

Starting point

)(GOE ˆ1

ˆ det ˆ det

kkOjjj OzOz

l

GOE characteristic polynomial)( kk

Nagao-Nishigaki (2001), Borodin-Strahov (2005)

}),({2 zzlcancellationReduced integral representation

}),({pf 2

erfc 22

2/)(

0}Im{ 1

2, ,

2 2

ll

l

l

zzKe

i

zzZdAp N

zzpp

Z ppNkN

pp ll

l22

2

}),({pf }),({

1

zzKzz N



» Sketch of derivation: I. Integrating out j ‘ s

Reduced integral representation

TSD

ISK

–part of a GOE matrix kernelD 22

}),({pf 2

erfc 22

2/)(

0}Im{ 1

2, ,

2 2

ll

l

l

zzKe

i

zzZdAp N

zzpp

Z ppNkN

pp

GOE skew-orthogonal polynomials

How do we calculate the integral ?..

not a projection operator !

DysonIntegrationTheorem

Inapplicable !!



» Sketch of derivation: I. Integrating out j ‘ s



» Outline reminder










» Sketch of derivation: II. Pfaffian integration theorem

Two fairly compact proofs




}),({pf 2

erfc 22

2/)(

0}Im{ 1

2, ,

2 2

ll

l

l

zzKe

i

zzZdAp N

zzpp

Z ppNkN

pp

Apply !!

jNjx

)1]2/[ ,0( ˆ tr

a nonuniversal ingredient

),,( ! 1)1(

,, llZ

lxx

pp NN

kN a probability to have all eigenvalues real

Zonal polynomials

l2kN4/)1(

, 2 NNNNp

ˆ 1̂ det )( ,2,

]2/[

0

zppzzG NNNN

N

N ll

l

Solved !!







Fredholm Pfaffian (Rains 2000)

Applied Mathematics



05




» Outline reminder







Statistics of real eigenvalues in GinOE

Exact formula for the distribution of the number k of real eigenvalues in the spectrum of n × n random Gaussian real (asymmetric) matrix Solution highlights a link between integrable structure of GinOE and the theory of symmetric functions

Even simpler solution is found for the entire generating function of the distribution of k

Pfaffian Integration Theorem as an extension of the Dyson Theorem (far beyond the present context)



» Conclusions

0 2 4 6 8 10 121. 1010

1. 108

1. 106

0.0001

0.01

Asymptotic analysis of the distribution of k (matrix size n taken to infinity)

Further extension of the Pfaffian integration theorem to determine all partial correlation functions

Looking for specific physical applications(weak non-Hermiticity)

!<< 1

GinOE modelASL HghHH

~ 1

directed chaos

?

Applied Mathematics

work in progress


in GinOE Spectra [ ]» What is next ?

Asymptotic analysis of the distribution of k (when k scales with E[k] and the matrix size n that is taken to infinity)

Applied Mathematics

Statistics of Real Eigenvaluesin GinOE Spectra

01

Statistics of Real Eigenvalues in GinOE Spectra

Eugene KanzieperDepartment of Applied

MathematicsH.I.T. - Holon Institute of

TechnologyHolon 58102, Israel

Gernot Akemann (Brunel)

Phys. Rev. Lett. 95, 230501 (2005)arXiv: math-ph/0703019 (J. Stat. Phys.)

in preparation

Alexei Borodin(Caltech)

[ ]

Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007

statistics of real eigenvalues in ginoe spectra snowbird conference on random matrix theory &...

Documents

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n n random real matrix

problem slide

timme et

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dyson integration theorem

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