On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles

Abstract

We consider the family of N-dimensional real symmetric matrices H with random independent entries whose variance is determined by a function U((x-y)/b). In the limit of (relatively) narrow band width 1<<b<<N, we obtain explicitly first terms of 1/b-expansion of the resolvent of H. The expressions derived show that the rate of decay of U(t) determines several scale of the universal form of the eigenvalue correlation function. In particular, in the case of U(t) = o(1/t^3), the Altshuller-Shklovski asymptotics with the ratio N/b^2 is obtained.