A Spectral Dominance Approach to Large Random Matrices

TL;DR

This paper introduces a spectral dominance method to analyze the limit spectrum of large random matrices, providing a new way to determine spectral measures via viscosity solutions of integro-differential equations, applicable to various models.

Contribution

It proposes a novel spectral dominance approach that simplifies the characterization of the limit spectrum of large random matrices using viscosity solutions.

Findings

Limit spectral measure derived from viscosity solutions.

Applicable to Dyson Brownian motions and Wishart processes.

Provides concise proofs for convergence of spectral measures.

Abstract

This paper presents a novel approach to characterize the dynamics of the limit spectrum of large random matrices. This approach is based upon the notion we call "spectral dominance". In particular, we show that the limit spectral measure can be determined as the derivative of the unique viscosity solution of a partial integro-differential equation. This also allows to make general and "short" proofs for the convergence problem. We treat the cases of Dyson Brownian motions, Wishart processes and present a general class of models for which this characterization holds.