Mesoscopic eigenvalue statistics for Wigner-type matrices

TL;DR

This paper establishes a universal central limit theorem for mesoscopic eigenvalue statistics of Wigner-type matrices, using an optimal local law for two-point functions and resolvent analysis.

Contribution

It introduces a new local law for two-point functions and extends mesoscopic CLT results to Wigner-type matrices with smooth test functions.

Findings

Proves a universal mesoscopic CLT for Wigner-type matrices.

Develops an optimal local law for two-point functions.

Extends eigenvalue statistics understanding in random matrix theory.

Abstract

We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions. The main novel ingredient is an optimal local law for the two-point function $T(z,\zeta)$ and a general class of related quantities involving two resolvents at nearby spectral parameters.