High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm

Abstract

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.

arXiv commentThis version: misprints corrected, some parts of the proofs simplified, general presentation improved. The final version to appear in: Random Operators and Stoch. Equations

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