Universality for certain Hermitian Wigner Matrices under weak moment conditions

TL;DR

This paper proves that the local eigenvalue statistics of certain Hermitian Wigner matrices follow universal patterns at the edge and in the bulk, under minimal moment conditions, extending universality results.

Contribution

It establishes universality for Gaussian divisible Hermitian Wigner matrices under optimal moment conditions, including only finite second moments for bulk universality.

Findings

Tracy-Widom universality at the spectral edge under fourth moment bound.

Bulk universality with only finite second moments.

Extension of universality results to broader classes of Wigner matrices.

Abstract

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.