# Symmetric elliptic functions, IRF models, and dynamic exclusion processes

## TL;DR

This paper introduces stochastic IRF models linked to elliptic quantum groups, explores their degenerations to dynamic particle systems, and analyzes their observables and asymptotics using novel symmetric elliptic functions.

## Contribution

It develops a new class of symmetric elliptic functions and applies them to analyze stochastic IRF models and their limits, revealing new integrable dynamic particle systems.

## Key findings

- Evaluation of observables for stochastic IRF models.
- Derivation of dynamic ASEP and SSEP models as limits.
- Asymptotic analysis of the dynamic SSEP.

## Abstract

We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$. For stochasic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of $q$-moments of the height function for the stochastic (higher spin) six vertex models. In a certain limit, the stochastic IRF models degenerate to (1+1)d interacting particle systems that we call dynamic ASEP and SSEP; their jump rates depend on local values of the height function. For the step initial condition, we evaluate averages of observables for them as well, and use those to investigate one-point asymptotics of the dynamic SSEP. The construction and proofs are based on remarkable properties (branching and Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric elliptic functions that arise as matrix elements in an infinite volume limit of the algebraic Bethe ansatz for $E_{\tau,\eta}(sl_2)$.