Free Convolution and Generalized Dyson Brownian Motion

TL;DR

This paper interprets free convolution of large random matrices as a generalized Dyson Brownian motion, revealing how eigenvalues evolve dynamically and identifying conditions for outliers outside the spectrum.

Contribution

It provides a dynamical particle interpretation of free convolution and extends Dyson Brownian motion to include generalized interactions and matrix products.

Findings

Eigenvalue evolution described by a generalized Dyson Brownian motion.

Outliers are signaled by divergence in the particle velocity.

Extension of results to products of free matrices.

Abstract

The eigenvalue spectrum of the sum of large random matrices that are mutually "free", i.e., randomly rotated, can be obtained using the formalism of R-transforms, with many applications in different fields. We provide a direct interpretation of the otherwise abstract additivity property of R-transforms for the sum in terms of a dynamical evolution of "particles" (the eigenvalues), interacting through two-body and higher-body forces and subject to a Gaussian noise, generalizing the usual Dyson Brownian motion with Coulomb interaction. Interestingly, the appearance of an outlier outside of the bulk of the spectrum is signalled by a divergence of the "velocity" of the generalized Dyson motion. We extend our result to products of free matrices.