Download Talk at NACE 2005

[
Applied Mathematics
Statistics of Real Eigenvalues
in GinOE Spectra
]
Statistics of Real Eigenvalues
in GinOE Spectra
Eugene Kanzieper
Gernot Akemann
Department of Applied Mathematics
H.I.T. - Holon Institute of Technology
Holon 58102, Israel
(Brunel)
Phys. Rev. Lett. 95, 230501 (2005)
arXiv: math-ph/0703019 (J. Stat. Phys.)
Alexei Borodin
(Caltech)
in preparation
Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007
42
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» The Problem
2005
2007
What is the probabilityStatistics
that an n × of
n random real matrix
with Gaussian i.i.d. entries has exactly k real eigenvalues?
Spectra
A.Complex
Edelman (mid-nineties)
41
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
40
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
39
» Ginibre’s random matrices: also physics
P
(H)DH  
 N / 2

exp  tr HH   dHij( q )
GinOE : H  R N N (   1)  GOE
GinSE : H  QN N (   4)  GSE
GinUE : H  CN N (   2)  GUE
N
q 1 i , j 1
1965
success
complexity

( )
N
2
Dropped Hermiticity…
Statistics of
Is
there any
Complex
Spectra
physics
?
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Ginibre’s random matrices: also physics
Is there any
physics
?
38
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
» Ginibre’s random matrices: also 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)
]
37
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Ginibre’s random matrices: also physics
directed chaos
<< 1
?
Is there any
HL  HS  h g H A
physics
GinOE model
~1
36
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
35
» Ginibre’s random matrices: also physics
Asymmetric L-R
Cross-Correlation
Matrices
Universal noise dressing
is still there !
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Ginibre’s random matrices: also 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)
Back to
1965 and Ginibre’s maths curiosity…
34
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline reminder
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
33
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Spectra of Ginibre’s random matrices

( )
N
P
(H)DH  
(almost) uniform
distribution
 N / 2
2

exp  tr HH   dHij( q )
N
q 1 i , j 1
depletion from
real axis
1965
accumulation along
real axis
32
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Spectra of Ginibre’s random matrices
1965
(almost) uniform
distribution
depletion from
real axis
accumulation along
real axis
31
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Spectra of Ginibre’s random matrices

N
PN z1 ,..., z N   C2 ( N )  zk1  zk2
k1 k2
2 N
 zk zk
e

1965
k 1
GinUE : jpdf + correlations
(almost) uniform
distribution
30
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Spectra of Ginibre’s random matrices

N
PN z1 ,..., zN   C4 ( N )  zk1  zk2 zk1  zk2
k1 k2
2
2 N
 zk zk
z

z
e
 k k
k 1
2
1965
Mehta, Srivastava 1966
GinUE : jpdf + correlations
GinSE : jpdf + correlations
depletion from
real axis
29
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Spectra of Ginibre’s random matrices

N
PH T ( N / N ) ( w 1 ,, w N )    j1   j2
j1  j2
N
e
j 1
  j2 / 2
1965
Mehta, Srivastava 1966
GinUE : jpdf + correlations
GinSE : jpdf + correlations
accumulation along
real axis
28
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Spectra of Ginibre’s random matrices

NN
N
k j10
1 j2
j 1
PHHTT((NN/)N()w( w
, , w )  
PH Tj1( N/ k) (j2w 1
,e, w N )
1 ,1 , w N N)  
  j2 / 2
?
1965
Key Feature H  T (N )
T (N )
…
N
accumulation along
T ( Nreal
) axis T ( N
k 10

/ k)
T ( N / 0) ... T ( N / k ) ... T ( N / N )
number of real eigenvalues
27
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline reminder
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
26
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
25
]
» Overview of major developments since 1965
Ginibre
1965

Lehmann & Sommers
Edelman, Kostlan & Shub
Edelman
1994
1997
1991
quarter of a century !!



NN

2n
1
1




E
[
k
]

k
p

1

O

)   , wPNH,kT{( N /,
PHT ( N / N ) ({wP})
H T ( N ) ( w 1 ,  , w N
k ) ( w 1 , n, w N )}
w   w 1k,
0
N
1  ,  k , z 1 , z 1 ,, z l , z l2

 
k 0
1
k real eigenvalues l pairs of c.c. eigenvalues
Correlation
Functions ?!
N
PH T ( N / k ) ({w})  C N , k  wi  w j
i j
 N w 2

 e j erfc  w j  w j


2
j 1


1/ 2




[
Applied Mathematics
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Overview of major developments since 1965
Ginibre
Lehmann & Sommers
1965

Edelman, Kostlan & Shub
Edelman
1994
1997
1991
quarter of a century !!



N
PH T ( N ) ( w 1 ,, w N )   PH T ( N / k ) ( w 1 ,, w N )
k 0
1
Correlation
Functions ?!
Borodin & Sinclair, arXiv: 0706.2670
Forrester & Nagao, arXiv: 0706.2020
Sommers, arXiv: 0706.1671
detailed k-th partial correlation functions are not available…
24
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline reminder
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
23
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Real vs complex eigenvalues
Edelman
N
p N ,k    dw j PH T ( N / k ) ({w})
1997
j 1
Probability to have all eigenvalues real
pN ,N  2 N ( N 1) / 4
(the smallest one)
Theorem
pN ,k  rN ,k  sN ,k 2

( rN ,k
& s N ,k
rational)


22
Applied Mathematics
[
]
Statistics of Real Eigenvalues
in GinOE Spectra
» Real vs complex eigenvalues
p N ,k
l
2
2
cN ,k  k
 zp  zp 
 z p  z p  ( zEdelman
N
 j2 / 2
2
p zp ) / 2

d Z p 
 erfc 
 e
 d j e


k!l ! pj 1 
2
i
i
2
1
 ({w})



dw
Im{
Z }0 pP
N ,k
k

j 1
  i   j
i j
N  k  2l
j
l
z
pq
p
H T ( N / k )
 zq
2
z p  zq
2
 
k
1997
l
j 1 p 1
j
 z p  j  z p 
wp  w 1,
,2
wNN(N{1) 1/ ,4,  k , z 1, z 1,, z l , z l }

 
pN,k
rN ,k
N ,N
k real eigenvalues l pairs of c.c. eigenvalues
Solved ?..

 s N ,k 2
+
21
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Real vs complex eigenvalues
MATHEMATICA
code up to
N 9
No Closed
Formula for
p N ,k
pN , N  2
 N ( N 1) / 4
20
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline reminder
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
19
[
Applied Mathematics
Statistics of Real Eigenvalues
in GinOE Spectra
]
18
» Probability to find exactly k real eigenvalues

The Answer
a probability to have all eigenvalues real
pN ,k  pN , N Fl ( x1 ,, xl )
N  k  2l
universal multivariate polynomials
j
1 l 2xl j  z p  z p  x j z ptr( 0z,p[ N / 2](z1) zˆ) / 2j
c N ,k
l  / 2
Fl p(Nx,k1 ,, xl ) ( d1)j e  d Z p 
 erfc 
 e


! 1 l j  2i 
k!l !  j 1
{l } jIm{
1 Z
 i 2 
}0j p
 k
p (l )
g
2
j
2
p
2
p
a nonuniversal ingredient
k

integer partitions

i   j



i j
l
z
pq

p
 zq
Even Better



l  l 1 1 , l 2 2 ,, l gGg N ( z ) 
Starting point
2
z p  zq
[ N / 2]
2
1
Fl l ( x1 ,...,
x
)

l   zZ(1l ) ( x1 ,..., xl )

z

j
p
j l !p
j 1 p 1
k


l
z
p N , N det
1̂  z ˆ
polynomials
 pN , N 2l zonal
l 0
Jack polynomials at α=2
Applied Mathematics
[
17
]
Statistics of Real Eigenvalues
in GinOE Spectra
» Probability to find exactly k real eigenvalues
0.01
0.0001
1.
10 6
1.
10 8
1.
10 10
0
2
4
6
8
10
12
8
10
12
0.4
0.3
0.2
0.1
0
0
2
4
6
No visible discrepancies with numeric simulations over 10 orders of magnitude !!
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
16
» Sketch of derivation: I. Integrating out j ‘ s

cancellation
Reduced
integral representation
 ({z, z})
2l
1
l 
pf K N ({z, z})
 z p  z p  ( z p2  z p2 ) / 2
2l 22l
( k kK
) characteristic
GOE pf
polynomial
 2l ({pzN,,kz})
 e
 A N ,l   d Z p erfc 
N ({ z , z }) 2l 2l
 i 2 
Im{ Z } 0 p 1
l
 det  z
Nagao-Nishigaki (2001), Borodin-Strahov (2005)

j 1
Starting point
p N ,k
j
 
 Oˆ det z j  Oˆ
l
cN ,k  k
 zp  zp 
 z p  z p  ( z p2  z p2 ) / 2
 j2 / 2
2

d

e
d
Z
erfc



 e


j
p 



k!l !  j 1
 2i 
 i 2 
Im{ Z }0 p 1
k
  i   j
i j
l
z
pq
p
 zq
2
z p  zq
2
 
k
l
j 1 p 1
j
 z p  j  z p 

Oˆ  GOE( k k )
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
15
» Sketch of derivation: I. Integrating out j ‘ s

Reduced integral representation
pN ,k  A N ,l
S I 
K
T
D S 
 z p  z p  ( z p2  z p2 ) / 2
 e
d Z p erfc 
pf K N ({z, z}) 2l 2l


 i 2 
Im{ Z } 0 p 1
l
2
D –part of a GOE 2  2 matrix kernel
GOE skew-orthogonal polynomials
not a projection
operator !
Dyson
Integration
Theorem
Inapplicable !!
How do we calculate the integral ?..
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline reminder
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
14
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem
Two fairly
compact proofs
13
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem
p N ,k  A N ,l
 z p  z p  ( z p2  z p2 ) / 2
 e
d Z p erfc 
pf K N ({z, z}) 2l 2l


 i 2 
Im{ Z } 0 p 1
l
2
12
[
Applied Mathematics
Statistics of Real Eigenvalues
in GinOE Spectra
11
]
» Sketch of derivation: II. Pfaffian integration theorem
pN ,k 
pN , N
l!
a probability to have all eigenvalues real
Z (1l ) ( x1 ,, xl )
N  k  2l
pN , N  2  N ( N 1) / 4
Zonal polynomials
GN ( z ) 
[ N / 2]
z
l 0
l

pN , N 2l  pN , N det 1̂  z ˆ

x j  tr (0, [ N / 2]1) ˆ
j
a nonuniversal ingredient
Solved !!
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem
10
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
09
» Sketch of derivation: II. Pfaffian integration theorem

Fredholm Pfaffian (Rains 2000)
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem



08
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem
07
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem
06
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Sketch of derivation: II. Pfaffian integration theorem
05
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Outline reminder
 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 ?
• Probability to find exactly k real eigenvalues and
inapplicability of the Dyson integration theorem
• Pfaffian integration theorem
 Conclusions & What is next ?
04
[
Applied Mathematics
Statistics of Real Eigenvalues
in GinOE Spectra
]
» Conclusions
 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
0.01
0.0001
 Pfaffian Integration Theorem as an extension of the
1.
10 6
1.
10 8
Dyson Theorem (far beyond the present context)
1.
10 10
0
2
4
6
8
10
12
03
Applied Mathematics
[
Statistics of Real Eigenvalues
in GinOE Spectra
]
» What is next ?

Looking for specific
physical applications
(weak non-Hermiticity)
!
directed chaos
<< 1
HL  HS  h g H A
GinOE model
~1
?
 Asymptotic analysis of the distribution of k (matrix size n
taken to infinity)
work in progress
 Asymptotic analysis of the distribution
of k (when k scales with E[k] and the matrix size n that is
taken to infinity)
 Further extension of the Pfaffian integration theorem to
determine all partial correlation functions
02
[
Applied Mathematics
Statistics of Real Eigenvalues
in GinOE Spectra
]
Statistics of Real Eigenvalues
in GinOE Spectra
Eugene Kanzieper
Gernot Akemann
Department of Applied Mathematics
H.I.T. - Holon Institute of Technology
Holon 58102, Israel
(Brunel)
Phys. Rev. Lett. 95, 230501 (2005)
arXiv: math-ph/0703019 (J. Stat. Phys.)
Alexei Borodin
(Caltech)
in preparation
Snowbird Conference on Random Matrix Theory & Integrable Systems, June 25, 2007
01