The combinatorics of hopping particles and positivity in Markov chains

2022/02/01 by Lauren Williams, Williams, Lauren K.
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2202.00214

openalex publication_date 2022/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. In this article I give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics (tableaux and multiline queues) and special functions (Askey-Wilson polynomials, Macdonald polynomials, and Schubert polynomials). I also make some general observations about positivity in Markov chains.

Related

• Tableaux combinatorics for the asymmetric exclusion process
• A Markov chain on permutations which projects to the PASEP
• From multiline queues to Macdonald polynomials via the exclusion process
• Matrix ansatz and combinatorics of the k-species PASEP
• The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model