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

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.

version 2: minor changes; formulas (1.4) and (2.2) corrected