129 research outputs found
Elliptic Determinantal Process of Type A
We introduce an elliptic extension of Dyson's Brownian motion model, which is
a temporally inhomogeneous diffusion process of noncolliding particles defined
on a circle. Using elliptic determinant evaluations related to the reduced
affine root system of types , we give determinantal martingale
representation (DMR) for the process, when it is started at the configuration
with equidistant spacing on the circle. DMR proves that the process is
determinantal and the spatio-temporal correlation kernel is determined. By
taking temporally homogeneous limits of the present elliptic determinantal
process, trigonometric and hyperbolic versions of noncolliding diffusion
processes are studied.Comment: v5: AMS-LaTeX, 35 pages, no figure, references updated for
publication in Probab. Theory Relat. Field
Elliptic Determinantal Processes and Elliptic Dyson Models
We introduce seven families of stochastic systems of interacting particles in
one-dimension corresponding to the seven families of irreducible reduced affine
root systems. We prove that they are determinantal in the sense that all
spatio-temporal correlation functions are given by determinants controlled by a
single function called the spatio-temporal correlation kernel. For the four
families , , and , we identify the systems of
stochastic differential equations solved by these determinantal processes,
which will be regarded as the elliptic extensions of the Dyson model. Here we
use the notion of martingales in probability theory and the elliptic
determinant evaluations of the Macdonald denominators of irreducible reduced
affine root systems given by Rosengren and Schlosser
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