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

TL;DR

This paper establishes a connection between Dyson's Brownian motion model and Dunkl processes of type A, revealing how the intertwining operator influences symmetric polynomials and describing the freezing behavior in the strong coupling limit.

Contribution

It demonstrates the equivalence between symmetric Dunkl processes and Dyson's model, and analyzes the effect of the intertwining operator on symmetric polynomials and the freezing regime.

Findings

Dunkl processes with symmetric initial conditions are equivalent to Dyson's model with beta=2k.

The intertwining operator maps symmetric polynomials to functions of their sum in the strong coupling limit.

Dyson's model freezes to a deterministic configuration related to Hermite polynomial roots in the strong coupling limit.

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.