  We consider an ensemble of nxn real symmetric random matrices A whose entries
are determined by independent identically distributed random variables that
have symmetric probability distribution. Assuming that the moment 12+2delta of
these random variables exists, we prove that the probability distribution of
the spectral norm of A rescaled to n^{-2/3} is bounded by a universal
expression. The proof is based on the completed and modified version of the
approach proposed and developed by Ya. Sinai and A. Soshnikov to study high
moments of Wigner random matrices.
