Even Walks and Estimates of High Moments of Large Wigner Random Matrices

We revisit the problem of estimates of moments E Tr(An) 2s of random n × n matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given by the structure of closed even walks w2s and their graphs g(w2s) that arise in these studies. We show that the total degree of a vertex α of g(w2s) depends not only on the self-intersections degree of α but also on the total number of all non-closed instants of self-intersections of w2s. This result is used to fill the gaps of earlier considerations. Here we are restricted to the simplest case of bounded random variables √ n(An)ij and of the asymptotic regime s = O(n ), δ > 0 as n → ∞.