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.Comment: version 2: minor changes; formulas (1.4) and (2.2) correcte