High moments of large Wigner random matrices and asymptotic properties of the spectral norm

We consider the Wigner ensemble of n x n real symmetric random matrices A((n)) whose entries are determined by independent identically distributed random variables {a(ij), i <= j} that have symmetric probability distribution with variance v(2) and study the asymptotic behavior of the spectral norm parallel to A((n))parallel to as n -> infinity. We prove that if the moment E|a(ij)|(12+2 delta)(0) with any strictly positive delta(0) exists, then the probability P{parallel to A((n))parallel to > 2v(1 + xn(-2/3))}, x > 0, is bounded in the limit of infinite n by an expression that does not depend on the details of the probability distribution of a(ij). The proof is based on the completed and modified version of the approach developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.