Universal behavior of correlations between eigenvalues of random matrices

Abstract

Recently, the smoothed correlation between the density of eigenvalues of Hermitian random matrices was found to be universal, that is, independent of the probability distribution from which the random matrices are taken. We study this universal correlation numerically by ensemble averaging, using the Monte Carlo sampling method. Although the density of eigenvalues and the "bare" correlation between the density of eigenvalues are certainly not universal, we find that the smoothed correlation indeed shows a universal behavior. The orthogonal and the symplectic ensembles are also studied numerically. The smoothed correlation is shown to be universal in these cases. © 1995 The American Physical Society.