Friday, February 24, 2012

1202.5052 (Sergio Andraus et al.)

Dyson's Brownian Motion Model as a Special Case of Dunkl Processes and
Dunkl's Intertwining Operators

Sergio Andraus, Makoto Katori, Seiji Miyashita
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.
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