A Note on the Concentration of Spectral Measure of Wigner's Matrices

TL;DR

This paper extends concentration inequalities to the empirical absolute moments of Wigner matrices with ±1 entries, using advanced eigenvalue distribution analysis to include power functions beyond Lipschitz bounds.

Contribution

It introduces new concentration inequalities for moments of Wigner matrices, broadening the scope beyond previous Lipschitz-restricted results.

Findings

Derived concentration inequalities for empirical moments of Wigner matrices

Extended Talagrand's measure concentration tools to power functions

Utilized eigenvalue distribution analysis by Soshnikov and Sinai

Abstract

In this short note we derive concentration inequalities for the empirical absolute moments of square symmetric matrices with independent symmetrically distributed +/-1 entries. Most of the previous results of this type are limited to functions with bounded Lipschitz constant, and therefore exclude the moments from consideration. Based on the fine analysis of the distribution of the largest eigenvalues developed by Soshnikov and Sinai, we extend the measure concentration tools of Talagrand to encompass power functions.