Two limiting regimes of interacting Bessel processes

TL;DR

This paper analyzes the steady-state distributions and limiting behaviors of interacting Bessel processes in two regimes, revealing connections to random matrix eigenvalues and polynomial zeroes, using Dunkl operator theory.

Contribution

It introduces new results on the steady-state distributions and relaxation times of interacting Bessel processes in large parameter regimes, utilizing Dunkl operators and deriving a novel expression for the intertwining operator.

Findings

Steady-state distribution for large beta matches beta-Laguerre eigenvalue distribution.

In the limit nu->infinity, particles converge to a single point.

Final particle positions in the beta->infinity regime align with Laguerre polynomial zeroes.

Abstract

We consider the interacting Bessel processes, a family of multiple-particle systems in one dimension where particles evolve as individual Bessel processes and repel each other via a log-potential. We consider two limiting regimes for this family on its two main parameters: the inverse temperature beta and the Bessel index nu. We obtain the time-scaled steady-state distributions of the processes for the cases where beta or nu are large but finite. In particular, for large beta we show that the steady-state distribution of the system corresponds to the eigenvalue distribution of the beta-Laguerre ensembles of random matrices. We also estimate the relaxation time to the steady state in both cases. We find that in the freezing regime beta->infinity, the scaled final positions of the particles are locked at the square root of the zeroes of the Laguerre polynomial of parameter nu-1/2 for any initial configuration, while in the regime nu->infinity, we prove that the scaled final positions of the particles converge to a single point. In order to obtain our results, we use the theory of Dunkl operators, in particular the intertwining operator of type B. We derive a previously unknown expression for this operator and study its behaviour in both limiting regimes. By using these limiting forms of the intertwining operator, we derive the steady-state distributions, the estimations of the relaxation times and the limiting behaviour of the processes.