High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm

TL;DR

This paper investigates the asymptotic behavior of the spectral norm of large Wigner random matrices, establishing universal bounds under certain moment conditions using advanced probabilistic techniques.

Contribution

It extends existing methods to prove universal bounds on the spectral norm distribution of Wigner matrices with finite high moments.

Findings

Spectral norm distribution is bounded by a universal expression when scaled by n^{-2/3}.

The approach adapts Sinai and Soshnikov's method for high moments analysis.

Results hold under the assumption of finite 12+2δ moments of matrix entries.

Abstract

We consider an ensemble of nxn real symmetric random matrices A whose entries are determined by independent identically distributed random variables that have symmetric probability distribution. Assuming that the moment 12+2delta of these random variables exists, we prove that the probability distribution of the spectral norm of A rescaled to n^{-2/3} is bounded by a universal expression. The proof is based on the completed and modified version of the approach proposed and developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.