Gap Universality of Generalized Wigner and beta-Ensembles

TL;DR

This paper proves that the distribution of individual eigenvalue gaps in the bulk of generalized Wigner and beta-ensembles is universal, extending previous results that required local averaging.

Contribution

It establishes the universality of single eigenvalue gap distributions in the bulk for generalized Wigner and beta-ensembles without the need for local averaging.

Findings

Single eigenvalue gap distributions are universal in the bulk.

Universality holds for all beta > 1 with analytic potentials.

Extends previous results from local averages to individual gaps.

Abstract

We consider generalized Wigner ensembles and general beta-ensembles with analytic potentials for any beta larger than 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian beta-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal.