An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

TL;DR

This paper surveys recent progress on universality limits for correlation functions in unitary ensembles, focusing on specific kernels and settings, and highlights open problems in the field.

Contribution

It provides an updated overview of universality limits for correlation functions in unitary ensembles, emphasizing specific kernels and excluding broader classes like $eta$ ensembles.

Findings

Universality limits hold for Airy, Bessel, and Sine kernels in certain settings.

The survey covers measures on compact intervals and exponential weights.

Open problems in the area are identified and discussed.

Abstract

We survey the current status of universality limits for $m$-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider $\beta$ ensembles for $\beta \neq 2$, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.