1. Random Matrices and Replica Trick Alex Kamenev Department of Physics, University of Minnesota

2. Random matrix,, is a Hamiltonian: partition function annealed average quenched average

3. Replica trick: n is integer (!); positive or negative Level statistics:

4. Averaging: anticommuting N - vector

5. Duality transformation:

6. Generalizations: H Q U U O Sp O 2. 3. Anderson localization (Schrodinger in random potential): NL  M

7. Saddle points:

8. Saddle point manifolds: 2. where

9. Analytical continuation: semicircle 1/N oscillations do not contribute

10. However: diverges for any non-integer n ! One needs a way to make sense of this series for |  | 1 ).

11. 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

12. Generalizations (continued): 4. Other symmetry classes: 5. Non—Hermitian random matrices. 6.Painleve method of analytical continuation (unitary classes). 7. Hard-core 1d bosons:

13. class G G 1 U(g) A (CUE)U(N)1g AI (COE)U(N)O(N)gTggTg AII (CSE)U(2N)Sp(2N)g T Jg AIII(chCUE)U(N+N’)U(N)*U(n’)Ig + Ig BD1(chCOE)SO(N+N’)SO(N)*SO(N’)Ig T Ig CII (chCSE)Sp(2N+2N’)Sp(2N)*Sp(2N’)Ig D Ig D, BSO(2N),SO(2N+1)1g CSp(2N)1g CISp(4N)U(2N)Ig + Ig DIII e/oSO(4N),SO(4N+2)U(2N),U(2N+1)g D g g 2 G; U(g) 2 G/G 1 ;

15. Calogero-Sutherland-Moser models: Van-der-Monde determinant

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

17. Painleve approach (unitary ensembles): E. Kanzieper 02 Hankel determinants Painleve IV transcendent