Physics – Mathematical Physics
Scientific paper
2012-02-22
Physics
Mathematical Physics
40 pages, 1 figure
Scientific paper
Dyson's Brownian motion model is a one-parameter family of log-potential interacting particle systems in one dimension parametrized by an inverse temperature 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. Dunkl processes are mathematically defined using differential- difference operators (Dunkl operators) associated with finite abstract vector sets called root systems. When the root system is specified to be of type A, Dunkl processes constitute a one-parameter family of interacting particles in one dimension, in which particles perform not only diffusive motion and mutual repulsion but also interchange positions spontaneously. In the present paper, 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 inverse temperature 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 A-type intertwining operator on symmetric polynomials from these processes' transition probability densities. In the zero-temperature limit, the intertwining operator maps all symmetric polynomials onto a function of the sum of their variables. This allows for the analysis of the zero-temperature limit of Dyson's model, which becomes a deterministic process with a final configuration proportional to a vector of the roots of the Hermite polynomials multiplied by the square root of the process time, while being independent of the initial configuration.
Andraus Sergio
Katori Makoto
Miyashita Seiji
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