Some refinements of Wigner’s semi‐circle law for Gaussian random matrices using superanalysis

Abstract

In Random Matrix Theory (R.M.T.), Wigner’s semi‐circle law gives a well‐known corner‐stone. We report mathematical refinements of this law as an application of superanalysis. That is, using Efetov’s idea, we rewrite the average of the empirical measure of the eigenvalue distribution of the Hermitian Gaussian matrices (GUE) in a compact form. Careful calculations give not only the precise convergence rate of that law, but also the precise rate of the edge mobility.