Single eigenvalue fluctuations of general Wigner-type matrices

TL;DR

This paper proves that individual eigenvalues of general Wigner-type matrices fluctuate in a Gaussian manner with universal variance, extending known results to broader matrix classes under a one-cut density assumption.

Contribution

It introduces a dynamical approach to mesoscopic spectral statistics, establishing a central limit theorem for eigenvalue fluctuations in Wigner-type matrices.

Findings

Eigenvalue fluctuations are Gaussian with universal variance.

The variance matches GOE/GUE cases for certain test functions.

Method applies to matrices with a one-cut density of states.

Abstract

We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain classes of test functions, the leading-order contribution to the variance agrees with the GOE/GUE cases.