Universality for Random Matrix Flows with Time-dependent Density

TL;DR

This paper proves that Dyson Brownian Motion leads to universal spectral behavior in a broad class of random matrices after a short time, under certain conditions, extending universality results to more general ensembles.

Contribution

It establishes bulk spectral universality for a wide class of Wigner-like matrices with time-dependent densities, under local rigidity and level repulsion assumptions.

Findings

Bulk spectral universality holds for these matrices after short evolution time.

Universality applies to deformed Wigner ensembles and matrices with non-stochastic variance.

Conditions like local rigidity and level repulsion are verified for these ensembles.

Abstract

We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from the Wigner semicircle law.