# Comparison theorem for some extremal eigenvalue statistics

## TL;DR

This paper presents a comparison theorem for extremal eigenvalue statistics of random matrices, enabling the transfer of known results across different ensembles under moment-matching conditions.

## Contribution

It introduces a new comparison method for extremal eigenvalue statistics, extending known eigenvalue gap results from GUE to broader classes of Hermitian matrices.

## Key findings

- Established a comparison theorem for extremal eigenvalue statistics
- Extended eigenvalue gap limit results from GUE to generalized Wigner matrices
- Validated the method for matrices with matching first four moments

## Abstract

We introduce a method for the comparison of some extremal eigenvalue statistics of random matrices. For example, it allows one to compare the maximal eigenvalue gap in the bulk of two generalized Wigner ensembles, provided that the first four moments of their matrix entries match. As an application, we extend results of Bourgade--Ben Arous and Feng--Wei that identify the limit of the maximal eigenvalue gap in the bulk of the GUE to all complex Hermitian generalized Wigner matrices.