# Elliptic Determinantal Processes and Elliptic Dyson Models

## TL;DR

This paper introduces seven families of one-dimensional interacting particle systems linked to affine root systems, proving their determinantal nature and identifying elliptic Dyson models through elliptic determinants and martingale techniques.

## Contribution

It extends Dyson models to elliptic cases associated with affine root systems and establishes their determinantal structure using elliptic determinants and martingale methods.

## Key findings

- Seven families of stochastic particle systems are determinantal.
- Explicit elliptic Dyson models are identified for four root system families.
- Correlation functions are expressed as determinants via a single kernel.

## 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.