On the 1/N corrections to the Green functions of random matrices with independent entries

Abstract

We propose a general approach to the construction of 1/N corrections to the Green function GN(z) of the ensembles of random real-symmetric and Hermitian N*N matrices with independent entries Hk,l. By this approach we study the correlation function CN(z1,z2) of the normalized trace N-1TrGN assuming that the average of mod Hk,l5 is bounded. We found that to leading order CN(z1,z2)=N-2F(z1,z2), where F(z1,2) only depends on the second and the fourth moments of Hk,l. For the correlation function of the density of energy levels we obtain an expression which, in the scaling limit only depends on the second moment of Hk,l. This can be viewed as supporting the universality conjecture of random matrix theory.