Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

TL;DR

This paper investigates how the entries of regular functions of Wigner matrices fluctuate as the matrix size grows large, extending previous Gaussian results to broader conditions on matrix entries and test functions.

Contribution

It extends the analysis of fluctuations of matrix entries of functions of Wigner matrices beyond Gaussian ensembles to matrices with finite moments and smooth test functions.

Findings

Fluctuations are characterized under finite moment conditions.

Results apply to non-Gaussian Wigner matrices.

Conditions include finite fourth moment for off-diagonal entries.

Abstract

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.