Asymmetric Exclusion Model and Weighted Lattice Paths

TL;DR

This paper interprets the stationary state algebra of ASEP through combinatorial weighted lattice paths, enabling new analytical methods and revealing connections to ballot numbers and polymer models.

Contribution

It introduces novel lattice path representations and combinatorial techniques to analyze ASEP's normalization factor and correlation functions.

Findings

Normalisation factor expressed via Ballot numbers

Recurrence coefficients linked to Ballot numbers

Asymptotic behaviour analyzed through omega-expansion

Abstract

We show that the known matrix representations of the stationary state algebra of the Asymmetric Simple Exclusion Process (ASEP) can be interpreted combinatorially as various weighted lattice paths. This interpretation enables us to use the constant term method (CTM) and bijective combinatorial methods to express many forms of the ASEP normalisation factor in terms of Ballot numbers. One particular lattice path representation shows that the coefficients in the recurrence relation for the ASEP correlation functions are also Ballot numbers. Additionally, the CTM has a strong combinatorial connection which leads to a new ``canonical'' lattice path representation and to the ``omega-expansion'' which provides a uniform approach to computing the asymptotic behaviour in the various phases of the ASEP. The path representations enable the ASEP normalisation factor to be seen as the partition function of a more general polymer chain model having a two parameter interaction with a surface.