Small deviation estimates for the largest eigenvalue of Wigner matrices

TL;DR

This paper provides precise small deviation estimates for the largest eigenvalue of Wigner matrices, using Green function comparison, and extends understanding of eigenvalue fluctuations in random matrix theory.

Contribution

It introduces a novel Green function comparison method to derive small deviation estimates for the largest eigenvalue of Wigner matrices.

Findings

Established precise right-tail small deviation estimates

Obtained less precise left-tail estimates

Demonstrated the effectiveness of Green function comparison

Abstract

We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.