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