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

Marton Balazs, Ross Bowen

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
280 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)514-528
Number of pages14
JournalAnnales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Volume54
Issue number1
Early online date19 Feb 2018
DOIs
Publication statusPublished - Feb 2018

Keywords

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

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