The eigenvalue spectrum of a large symmetric random matrix with exponential distributed elements

Abstract

A replica solution for the averaged eigenvalue spectrum of a large symmetric N*N matrix with an exponential distribution p(Mij)=( square root N/M0ij)exp(- square root NMij/M0ij) (M0ij=M0) of the elements is presented. This problem is reduced to the solution for the averaged eigenvalue spectrum of a homogeneous matrix M0ij=M0 with an added Gaussian random matrix. The main part of the obtained spectral density is the well known semicircular law rho ( lambda )=(1/2 pi M02) square root 4M02- lambda 2. In contrast to the Gaussian random matrix an additional second spectral region in the vicinity of M0 square root N is observed. The analytic result is verified by numerically obtained spectra of such matrices.