PDE methods in random matrix theory

TL;DR

This paper explores how PDE techniques can be applied to compute eigenvalue distributions, called Brown measures, in random matrix theory, especially focusing on large-N limits and free probability models.

Contribution

It introduces PDE methods for calculating Brown measures in operator algebra frameworks, extending the analysis of eigenvalue distributions in random matrix models.

Findings

PDE methods effectively compute Brown measures for certain random matrix ensembles.

The circular law case is analyzed in detail using PDE techniques.

Recent work on free multiplicative Brownian motion is discussed with PDE applications.

Abstract

This article begins with a brief review of random matrix theory, followed by a discussion of how the large-$N$ limit of random matrix models can be realized using operator algebras. I then explain the notion of "Brown measure," which play the role of the eigenvalue distribution for operators in an operator algebra. I then show how methods of partial differential equations can be used to compute Brown measures. I consider in detail the case of the circular law and then discuss more briefly the case of the free multiplicative Brownian motion, which was worked out recently by the author with Driver and Kemp.