Weakly universal dynamical correlations between eigenvalues of large random matrices

TL;DR

This paper demonstrates that the dynamical correlations of eigenvalues in large random matrices, evolving via Dyson Brownian motion, maintain a universal form over time, extending the known static universality to dynamic settings.

Contribution

It extends the concept of universality in eigenvalue correlations from static matrices to those undergoing Dyson Brownian motion, revealing persistent universal behavior over time.

Findings

Dynamical correlations preserve universality over time.

Eigenvalue correlations follow a PDE in the complex plane.

Universal behavior is maintained during matrix evolution.

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.