Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime

Abstract

We consider a one-dimensional system of Brownian particles that repel each other through a logarithmic potential. We study two formulations for the system and the relation between them. The first, Dyson's Brownian motion model, has an interaction coupling constant determined by the parameter beta > 0. When beta = 1,2 and 4, this model can be regarded as a stochastic realization of the eigenvalue statistics of Gaussian random matrices. The second system comes from Dunkl processes, which are defined using differential-difference operators (Dunkl operators) associated with finite abstract vector sets called root systems. When the type-A root system is specified, Dunkl processes constitute a one-parameter system similar to Dyson's model, with the difference that its particles interchange positions spontaneously. We prove that the type-A Dunkl processes with parameter k > 0 starting from any symmetric initial configuration are equivalent to Dyson's model with the parameter beta = 2k. We focus on the intertwining operators, since they play a central role in the mathematical theory of Dunkl operators, but their general closed form is not yet known. Using the equivalence between symmetric Dunkl processes and Dyson's model, we extract the effect of the intertwining operator of type A on symmetric polynomials from these processes' transition probability densities. In the strong coupling limit, the intertwining operator maps all symmetric polynomials onto a function of the sum of their variables. In this limit, Dyson's model freezes, and it becomes a deterministic process with a final configuration proportional to the roots of the Hermite polynomials multiplied by the square root of the process time, while being independent of the initial configuration.

arXiv commentLaTeX, 30 pages, 1 figure, 1 table. Corrected for submission to Journal of Physics A

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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
• Two limiting regimes of interacting Bessel processes
• 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