Abstract

We consider the family of ensembles of random symmetric N × N matrices H (N,b) of the band-type structure with characteristic length b. The variance of the matrix entries H (N,b) (x, y) is proportional to u ( x−y b) with certain decaying function u(t) ≥ 0. In the limit of (relatively) narrow band width 1 ≪ b ≪ N, we derive explicit expressions for the first terms of 1/b-expansions of the average of the Green function N −1 Tr(H (N,b) −z) −1 and its correlation function as well. The expressions obtained show that there exist several scales of the universal forms of the spectral correlation function. These scales are determined by the rate of decrease of the function u(t). They coincide with those detected in theoretical physics for the localization length and density-density correlator in the band-type random matrix ensembles. 1 Problem, motivation and results

Citations