Interacting Particle Systems and Jacobi Style Identities

TL;DR

This paper introduces new Jacobi-style identities derived from analyzing interacting particle systems with up to two particles per site, connecting statistical mechanics models to combinatorial identities involving Frobenius partitions.

Contribution

It develops a framework for relating particle systems with product blocking measures to Jacobi identities, extending to k-exclusion processes and generalised Frobenius partitions.

Findings

Derived new three-variable Jacobi identities from particle systems.

Connected specific processes to classical identities like the Jacobi triple product.

Extended identities to k-exclusion processes with k-repetition conditions.

Abstract

We consider the family of nearest neighbour interacting particle systems on $\mathbb{Z}$ allowing $0$, $1$ or $2$ particles at a site. We parametrize a wide subfamily of processes exhibiting product blocking measure and show how this family can be "stood up" in the sense of Bal\'azs and Bowen (2018). By comparing measures we prove new three variable Jacobi style identities, related to counting certain generalised Frobenius partitions with a $2$-repetition condition. By specialising to specific processes we produce two variable identities that are shown to relate to Jacobi triple product and various other identities of combinatorial significance. The family of $k$-exclusion processes for arbitrary $k$ are also considered and are shown to give similar Jacobi style identities relating to counting generalised Frobenius partitions with a $k$-repetition condition.