Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 079, 36 pages      arXiv:1703.03914
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

Elliptic Determinantal Processes and Elliptic Dyson Models

Makoto Katori
Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Received April 19, 2017, in final form September 29, 2017; Published online October 04, 2017

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.

Key words: elliptic determinantal processes; elliptic Dyson models; determinantal martingales; elliptic determinant evaluations; irreducible reduced affine root systems.

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