Title: Elliptic Determinantal Processes and Elliptic Dyson Models,
Abstract: 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 ${A}_{N-1}$, ${B}_N$, ${C}_N$ and ${D}_N$, 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.