A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

TL;DR

This paper proves a Central Limit Theorem for the eigenvalue counting function of Wigner matrices, extending known results for GUE matrices to a broader class using advanced probabilistic and localization techniques.

Contribution

It establishes a CLT for eigenvalue counts in Wigner matrices, leveraging variance asymptotics and the Tao-Vu Four Moment Theorem for a wide class of matrices.

Findings

CLT holds for eigenvalue counting in Wigner matrices.

Variance asymptotics are crucial for the proof.

Extension from GUE to general Wigner matrices achieved.

Abstract

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin.