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.

18 pages, 1 figure

References in corpus (3)

• From duality to determinants for q-TASEP and ASEP
• Euler hydrodynamics of one-dimensional attractive particle systems
• Universal scaling for second class particles in a one-dimensional misanthrope process

Cited by in corpus (5)

• Dynamics of the box-ball system with random initial conditions via Pitman's transformation
• Mallows Product Measure
• Interacting Particle Systems and Jacobi Style Identities
• Second Class Particle Behaviour in Blocking ASEP
• Unifying particle-based and continuum models of hillslope evolution with a probabilistic scaling technique