Fluctuations of Functions of Wigner Matrices

TL;DR

This paper demonstrates that the matrix elements of functions of Wigner matrices exhibit fluctuations of order N^{-1/2} and identifies their limiting behavior, relaxing previous regularity constraints.

Contribution

It extends the understanding of fluctuations of matrix elements for functions of Wigner matrices under weaker regularity conditions.

Findings

Matrix elements fluctuate at scale N^{-1/2}.

Limiting fluctuation distribution identified.

Applicable to functions with bounded variation.

Abstract

We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation. Our result holds for any function $f$ of the matrix that has bounded variation and thus considerably relaxes the regularity requirement imposed in [7,11].