Quantitative gap universality for Wigner matrices

TL;DR

This paper establishes explicit convergence rates for eigenvalue gaps in generalized Wigner matrices, extending universality results to matrices with atomic entries and providing optimal eigenvalue separation.

Contribution

It introduces a method extending the 4 moment matching to arbitrary moments, enabling precise comparison of resolvents at very fine scales.

Findings

Convergence rate of $N^{-1/2 + \\epsilon}$ for eigenvalue gaps in the bulk.

Universality of smallest eigenvalue gaps for Hermitian matrices.

Optimal eigenvalue separation for discrete matrices with finitely supported entries.

Abstract

We obtain the explicit rate of convergence $N^{-1/2 + \epsilon}$ for the gaps of generalized Wigner matrices in the bulk of the spectrum, for distributions of matrix entries possibly atomic and supported on enough points. The proof proceeds by a Green function comparison, coupled with the relaxation estimate from [5]. In particular, we extend the 4 moment matching method [33] to arbitrary moments, allowing to compare resolvents down to the submicroscopic scale $N^{-3/2 + \epsilon}$. This method also gives universality of the smallest gaps between eigenvalues for the Hermitian symmetry class, providing a universal, optimal separation of eigenvalues for discrete random matrices with entries supported on $\Omega(1)$ points.