Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Márton Balázs and Ross Bowen

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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.

Résumé

Nous passons en revue l’existence de mesures bloquantes de forme produit dans le contexte général des processus misanthropes asymétriques, au plus proche voisin, en dimension 1. Cela recouvre les modèles dits d’exclusion, de << zero range >>, de << bricklayers >>, et bien d’autres. Nous caractérisons les cas où de telles mesures existent en volume infini, et les cas où des frontières doivent être ajoutées. En nous intéressant aux distances entre particules, nous étendons la construction à certains systèmes de particules à valeurs dans $\{0,1\}$ qui ne sont pas misanthropes, tels que le $q$-ASEP et le processus de Katz-Lebowitz-Spohn. Au passage, nous obtenons une décomposition ergodique des mesures bloquantes de forme produit en des composantes caractérisées par une quantité conservée non triviale. Une conséquence intéressante, dans le cas de l’exclusion simple et du processus << zero range >>, est que cela donne une preuve purement probabiliste du triple produit de Jacobi, une identité célèbre qui intervient en théorie des nombres et dans la combinatoire des partitions. De façon surprenante, dans notre contexte, cette formule découle très naturellement de la correspondance entre l’exclusion et le processus << zero range >>.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 514-528.

Dates
Received: 9 June 2016
Revised: 30 November 2016
Accepted: 7 December 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030837

Digital Object Identifier
doi:10.1214/16-AIHP813

Mathematical Reviews number (MathSciNet)
MR3765898

Zentralblatt MATH identifier
06880063

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Blocking measure Interacting particle systems Reversible stationary distribution Jacobi triple product

Citation

Balázs, Márton; Bowen, Ross. Product blocking measures and a particle system proof of the Jacobi triple product. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 514--528. doi:10.1214/16-AIHP813. https://projecteuclid.org/euclid.aihp/1519030837


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