Random Matrices and Replica Trick

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: 2. 3. Anderson localization (Schrodinger in random potential): NLsM

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 |z|<1, (but not for |z|>1 ) .

11. Generalizations: • Higher order correlators, , • (but exact for b=2). • 2. Other ensembles: b=1,2,4 (O,U,Sp) volume factors: for b=1,2 two manifolds p=0,1; for b=4 – three p=0,1,2. 3. Arbitrary b: Calogero-Sutherland-Moser models

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

13. g2G; U(g) 2 G/G1 ;

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

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

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