Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

TL;DR

This paper extends Dyson's Brownian motion models to hermitian matrix-valued processes, exploring their eigenvalue distributions across all Altland-Zirnbauer classes, and provides a stochastic calculus proof of the Harish-Chandra formula.

Contribution

It introduces new noncolliding diffusion systems for matrix eigenvalues, covering all symmetry classes, and links eigenvalue processes with noncolliding particle systems.

Findings

Realization of all ten Altland-Zirnbauer classes as eigenvalue distributions.

Introduction of one-parameter and two-parameter noncolliding Bessel and meander processes.

Stochastic calculus proof of the Harish-Chandra (Itzykson-Zuber) formula.

Abstract

As an extension of the theory of Dyson's Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor's generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish-Chandra (Itzykson-Zuber) formula of integral over unitary group is established.