Eigenvalue processes in light of Riemannian submersion and gradient flow of isospectral orbits

TL;DR

This paper unifies the understanding of eigenvalue processes in random matrix theory using Riemannian geometry, revealing geometric origins of drift and collision phenomena in eigenvalue dynamics.

Contribution

It demonstrates that eigenvalue processes arise from projecting Brownian motion through Riemannian submersions, linking geometric curvature to eigenvalue behavior, and extends to eigenvector and $eta$-Dyson processes.

Findings

Eigenvalue processes are projections of Brownian motion via Riemannian submersions.

Mean curvature contributes to drift and eigenvalue collisions.

Framework recovers eigenvector processes and general $eta$-Dyson Brownian motion.

Abstract

We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced by isometric action of compact Lie groups, whose orbits have nonzero mean curvature, which contributes to drift terms and is the log gradient of orbit volume function, showing in another way that eigenvalues collide whenever the fibre is degenerate. We thus provide a unified treatment and better connection between eigenvalue processes in different settings with the language of Riemannian geometry. Under such interpretation, we see how we can naturally recover eigenvector processes and derive $\beta$ process such as $\beta$-Dyson Brownian motion for general $\beta>0$. \textbf{KEYWORDS}: eigenvalue process, symmetric space, mean curvature flow, gradient flow, isospectral manifold, random matrix ensemble