Multiple-particle diffusion processes from the viewpoint of Dunkl operators: relaxation to the steady state

TL;DR

This paper studies the relaxation and steady-state behavior of multi-particle Dunkl processes, showing convergence to eigenvalue distributions of random matrix ensembles and fixed final positions related to polynomial zeros.

Contribution

It establishes finite-time convergence of scaled steady states to random matrix eigenvalue distributions and characterizes final positions in the freezing limit for Dunkl processes.

Findings

Scaled steady states converge to eigenvalue distributions of beta-Hermite and beta-Laguerre ensembles.

Final particle positions in the freezing limit are at zeros of Hermite and Laguerre polynomials.

Dunkl processes in general converge to a steady-state distribution depending on process type.

Abstract

Two families of stochastic interacting particle systems, the interacting Brownian motions and Bessel processes, are defined as extensions of Dyson's Brownian motion models and the eigenvalue processes of the Wishart and Laguerre processes where the parameter $\beta$ from random matrix theory is taken as a real positive number. These are systems where several particles evolve as individual Brownian motions and Bessel processes while repelling mutually through a logarithmic potential. These systems are also special cases of Dunkl processes, a broad family of multivariate stochastic processes defined by using Dunkl operators. In this thesis, the steady state under an appropriate scaling and and the freezing ($\beta\to\infty$) regime of the interacting Brownian motions and Bessel processes are studied, and it is proved that the scaled steady-state distributions of these processes converge in finite time to the eigenvalue distributions of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrices. It is also shown that the scaled final positions of the particles in these processes are fixed at the zeroes of the Hermite and Laguerre polynomials in the freezing limit. These results are obtained as the consequence of two more general results proved in this thesis. The first is that Dunkl processes in general converge in finite time to a scaled steady-state distribution that only depends on the type of Dunkl process considered. The second is that in the freezing limit, their scaled final position is fixed to a set of points called the peak set, which is the set of points which maximizes their steady-state distribution.