Universality limits via “old style” analysis

Summary

Techniques from “old style” orthogonal polynomials have turned out to be useful in establishing universality limits for fairly general measures. We survey some of these.

1. Introduction

We focus on the classical setting of random Hermitian matrices: consider a probability distribution P⁽ⁿ⁾ on the space of n by n Hermitian matrices M = (mij)_1≤i,j≤n:

Here w is some nonnegative function defined on Hermitian matrices, and c is a normalizing constant. The most important case is

for appropriate functions Q. In particular, the choice (Q(M) =M², leads to the Gaussian unitary ensemble (apart from scaling) that was considered by Wigner, in the context of scattering theory for heavy nuclei. When expressed in spectral form, that is as a probability density function on the eigenvalues x₁≤x₂≤...≤x_n of M, it takes the form

See [Deift 1999, p. 102 ff.]. Again, c is a normalizing constant. Note that w now can be any nonnegative measurable function.