Weakly universal dynamical correlations between eigenvalues of large random matrices

Abstract

It was shown roughly thirty years ago that the density correlations of eigenvalues of large random matrices display a universal form, independent of most of the details of the distribution of the random matrix itself. We show that when the matrix elements evolve according to a Dyson Brownian motion, dynamical correlations retain a large degree of the universality found at equal times when expressed in terms of the characteristics of some partial differential equation in the complex plane.