Fluctuations of eigenvalues of matrix models and their applications

TL;DR

This paper derives correction terms for the expected linear eigenvalue statistics in matrix models with any beta, and uses these to establish bulk universality for certain symmetric and symplectic models.

Contribution

It provides the first order correction terms for eigenvalue expectations and proves bulk universality for real symmetric and symplectic matrix models with a given potential.

Findings

First order correction terms for eigenvalue expectations

Bulk universality established for real symmetric models

Bulk universality established for symplectic models

Abstract

We study the expectation of linear eigenvalue statistics of matrix models with any $\beta>0$, assuming that the potential $V$ is a real analytic function and that the corresponding equilibrium measure has a one-interval support. We obtain the first order (with respect to $n^{-1}$) correction terms for the expectation and apply this result to prove bulk universality for real symmetric and symplectic matrix models with the same $V$.