Universality of random matrix dynamics

TL;DR

This paper demonstrates that local spectral statistics of random matrix evolution operators are highly universal, depending primarily on the width-to-spacing ratio, with implications for stochastic processes and matrix products.

Contribution

It introduces the concept of width-to-spacing ratio as a key universal parameter for local spectral statistics in random matrix dynamics.

Findings

Local spectral properties depend mainly on the width-to-spacing ratio.

Universal behavior observed across different stochastic processes.

Duality established between kernels for Brownian motion and Lyapunov matrices.

Abstract

We discuss the concept of width-to-spacing ratio which plays the central role in the description of local spectral statistics of evolution operators in multiplicative and additive stochastic processes for random matrices. We show that the local spectral properties are highly universal and depend on a single parameter being the width-to-spacing ratio. We discuss duality between the kernel for Dysonian Brownian motion and the kernel for the Lyapunov matrix for the product of Ginibre matrices.