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

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

2010-05-18

arxiv.org/abs/1005.3231v2

Mathematics

Probability

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

Scientific paper

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.

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