Edge fluctuations of eigenvalues of Wigner matrices

TL;DR

This paper establishes moderate deviation principles for eigenvalue counts and the largest eigenvalues of Wigner matrices near the spectrum edge, extending known asymptotics and employing advanced probabilistic techniques.

Contribution

It introduces a moderate deviation principle for eigenvalue fluctuations near the spectrum edge of Wigner matrices, extending results to a broad class of matrices using the Four Moment Theorem.

Findings

MDP for eigenvalue counts near the spectrum edge

MDP for the $i$th largest eigenvalue near the edge

Extension of results to various Wigner matrices

Abstract

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval close to the edge of the spectrum. Moreover we prove a MDP for the $i$th largest eigenvalue close to the edge. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson. The extension to large families of Wigner matrices is based on the Tao and Vu Four Moment Theorem. Possible extensions to other random matrix ensembles are commented.