Random Matrices and Replica Trick

Alex Kamenev

Department of Physics,

University of Minnesota

Random matrix, , is a Hamiltonian:

partition function

annealed average

quenched average

Replica trick:

n is integer (!); positive or negative

Level statistics:

Averaging:

anticommuting N - vector

Duality transformation:

Generalizations:

H Q

U U

O Sp

Sp O

2.

3. Anderson localization (Schrodinger in random potential):

NLM

Saddle points:

Saddle point manifolds:

2.

where

Analytical continuation:

semicircle 1/N oscillations do not contribute

However:

diverges for any non-integer n !

One needs a way to make sense of this series for ||<1, (but not for ||>1 ) .

Generalizations:

1. Higher order correlators, , (but exact for =2).

2. Other ensembles: =1,2,4 (O,U,Sp) volume factors:

for 1,2 two manifolds p=0,1; for 4 – three p=0,1,2.

3. Arbitrary Calogero-Sutherland-Moser models

Generalizations (continued):

4. Other symmetry classes:

5. Non—Hermitian random matrices.

6. Painleve method of analytical continuation (unitary classes).

7. Hard-core 1d bosons:

class G G1 U(g)

A (CUE) U(N) 1 g

AI (COE) U(N) O(N) gTg

AII (CSE) U(2N) Sp(2N) gTJg

AIII(chCUE) U(N+N’) U(N)*U(n’) Ig+Ig

BD1(chCOE) SO(N+N’) SO(N)*SO(N’) IgTIg

CII (chCSE) Sp(2N+2N’) Sp(2N)*Sp(2N’) IgD Ig

D, B SO(2N),SO(2N+1) 1 g

C Sp(2N) 1 g

CI Sp(4N) U(2N) Ig+Ig

DIII e/o SO(4N),SO(4N+2) U(2N),U(2N+1) gD g

g 2 G; U(g) 2 G/G1 ;

Calogero-Sutherland-Moser models:

Van-der-Monde determinant

where

Integral identity: Z. Yan 92; J. Kaneko 93

Painleve approach (unitary ensembles): E. Kanzieper 02

Hankel determinants

Painleve IV transcendent