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 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.

LaTeX, 33 pages, 3 figures, corrected for publication on J. Phys. A: Math. Theor

References in corpus (5)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Noncolliding Brownian Motion and Determinantal Processes
• Noncolliding Squared Bessel Processes
• Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime
• Elliptic Determinantal Process of Type A

Cited by in corpus (9)

• Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices
• Limit theorems for multivariate Bessel processes in the freezing regime
• Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
• Functional central limit theorems for multivariate Bessel processes in the freezing regime
• Two-step asymptotics of scaled Dunkl processes
• Limit theorems for Jacobi ensembles with large parameters
• Dunkl jump processes: relaxation and a phase transition
• On the differential equations of frozen Calogero-Moser-Sutherland particle models
• Multiple-particle diffusion processes from the viewpoint of Dunkl operators: relaxation to the steady state