On Universality for Orthogonal Ensembles of Random Matrices

TL;DR

This paper proves the universality of local eigenvalue statistics in the bulk spectrum for orthogonal invariant matrix models with real analytic potentials, using a novel representation of the differentiation operator matrix.

Contribution

It introduces a new approach by representing the differentiation operator matrix as a product of Toeplitz matrices, advancing understanding of eigenvalue universality in orthogonal ensembles.

Findings

Established universality in the bulk spectrum for orthogonal ensembles.

Connected the Tracy-Widom formula to the matrix representation approach.

Provided a new mathematical framework for analyzing eigenvalue statistics.

Abstract

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.