Two limiting regimes of interacting Bessel processes

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 β and the Bessel index ν. We obtain the time-scaled steady-state distributions of the processes for the cases where β or ν are large but finite. In particular, for large β we show that the steady-state distribution of the system corresponds to the eigenvalue distribution of the β-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 β → ∞, the scaled final positions of the particles are locked at the square root of the zeroes of the Laguerre polynomial of parameter ν − 1/2 for any initial configuration, while in the regime ν → ∞, 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.