Product blocking measures and a particle system proof of the Jacobi triple product

TL;DR

This paper explores product blocking measures in particle systems, extending their construction to new models, and provides a probabilistic proof of the Jacobi triple product using exclusion-zero range correspondence.

Contribution

It characterizes conditions for product blocking measures in infinite volume and extends the framework to new particle systems, offering a novel probabilistic proof of the Jacobi triple product.

Findings

Characterization of product blocking measures in various particle systems.

Extension of construction to 0-1 valued systems like q-ASEP.

Probabilistic proof of the Jacobi triple product from particle system correspondence.

Abstract

We review product form blocking measures in the general framework of nearest neighbor asymmetric one dimensional misanthrope processes. This class includes exclusion, zero range, bricklayers, and many other models. We characterize the cases when such measures exist in infinite volume, and when finite boundaries need to be added. By looking at inter-particle distances, we extend the construction to some 0-1 valued particle systems e.g., q-ASEP and the Katz-Lebowitz-Spohn process, even outside the misanthrope class. Along the way we provide a full ergodic decomposition of the product blocking measure into components that are characterized by a non-trivial conserved quantity. Substituting in simple exclusion and zero range has an interesting consequence: a purely probabilistic proof of the Jacobi triple product, a famous identity that mostly occurs in number theory and the combinatorics of partitions. Surprisingly, here it follows very naturally from the exclusion - zero range correspondence.