A class of even walks and divergence of high moments of large Wigner random matrices

TL;DR

This paper investigates the divergence of high moments in large Wigner matrices with heavy-tailed entries, proposing that finite twelfth moments are necessary for universal bounds on these moments.

Contribution

It introduces a new class of even walks to analyze moment divergence and hypothesizes the importance of the twelfth moment for universality in Wigner matrices.

Findings

Constructed subset W' of walks with divergent sums

Identified the threshold of moments proportional to n^{2/3}

Linked finiteness of the twelfth moment to universal bounds

Abstract

We study high moments of truncated Wigner nxn random matrices by using their representation as the sums over the set W of weighted even closed walks. We construct the subset W' of W such that the corresponding sum diverges in the limit of large n and the number of moments proportional to n^{2/3} for any truncation of the order n^{1/6+epsilon}, epsilon>0 provided the probability distribution of the matrix elements is such that its twelfth moment does not exist. This allows us to put forward a hypothesis that the finiteness of the twelfth moment represents the necessary condition for the universal upper bound of the high moments of large Wigner random matrices.