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

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.