Bulk Universality and Related Properties of Hermitian Matrix Models

TL;DR

This paper presents a new proof of bulk universality in Hermitian matrix models using orthogonal polynomial techniques without relying on asymptotics, and also strengthens results on eigenvalue distribution.

Contribution

It introduces a novel proof method for bulk universality that avoids orthogonal polynomial asymptotics and improves understanding of eigenvalue distribution properties.

Findings

Established bulk universality under weaker smoothness conditions on the potential.

Derived the sine kernel as a solution to a non-linear integro-differential equation.

Strengthened results on the existence and properties of the limiting eigenvalue distribution.

Abstract

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.