Theory of random matrices with strong level confinement

TL;DR

This paper develops a theory for large N random matrices with non-Gaussian distributions, deriving asymptotically exact spectral properties and revealing universal local correlations independent of the distribution.

Contribution

It introduces a novel approach using orthogonal polynomials with exponential weights to analyze non-Gaussian random matrix ensembles and connects polynomial asymptotics with Dyson's mean-field theory.

Findings

Local eigenvalue correlations are universal in the large-N limit.

Global spectral correlations depend only on spectrum endpoints.

Established a new link between Szeg"o function structure and Dyson's mean-field equation.

Abstract

Unitary ensembles of large N x N random matrices with a non-Gaussian probability distribution P[H] ~ exp{-TrV[H]} are studied using a theory of polynomials orthogonal with respect to exponential weights. Asymptotically exact expressions for density of levels, one- and two-point Green's functions are calculated. We show that in the large-N limit the properly rescaled local eigenvalue correlations are independent of P[H] while global smoothed connected correlations depend on P[H] only through the endpoints of spectrum. We also establish previously unknown intimate connection between structure of Szeg\"o function entering strong polynomial asymptotics and mean-field equation by Dyson.