Dyson Brownian Motion and motion by mean curvature

TL;DR

This paper constructs Dyson Brownian motion for all positive beta values by adapting geometric methods, revealing that eigenvalues repel Coulombically and group orbits follow mean curvature flow, unifying stochastic and geometric perspectives.

Contribution

It introduces a geometric construction of Dyson Brownian motion for all beta, linking eigenvalue dynamics to mean curvature flow of group orbits.

Findings

Eigenvalues exhibit Coulombic repulsion for infinite beta.

Group orbits evolve according to mean curvature flow.

Unified geometric framework for Dyson Brownian motion.

Abstract

We construct Dyson Brownian motion for $\beta \in (0,\infty]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When $\beta$ is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.