Asymptotic expansions for β matrix models and their applications to the universality conjecture

Summary

We consider _ matrix models with real analytic potentials for both one-cut and multi-cut regimes. We discuss recent results on the asymptotic expansion of the correlators and partition functions and their applications to the studies of random matrices.

1. Introduction

We consider the probability measure on ℝⁿ of the form

where the function H, which we call the Hamiltonian to stress the analogy with statistical mechanics, and the normalizing constant Q_n,β[V] (partition function) have the form

The function V (called the potential) is a real-valued Hölder function satisfying the condition

We will study the asymptotic behavior (for large n) of Q_n,β[V]and the marginal densities of (1-1) (correlation functions)

The distribution (1-1) can be considered for any β> 0, but the cases β = 1, 2, 4 are especially important, since they correspond to the eigenvalue distribution of real symmetric, hermitian, and symplectic matrix models respectively.