Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions

TL;DR

This paper explores the combinatorial relationships between different lattice path representations of the stationary state algebra in a simple asymmetric exclusion process, using bijections and involutions to connect them.

Contribution

It introduces new bijections and involutions that relate three different lattice path models of the stationary state algebra in the ASEP.

Findings

All three path sets are combinatorially related.

Bijections and involutions connect the different path representations.

The methods clarify the structure of the stationary state algebra.

Abstract

We study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process. Each of the representations considered correspond to a different set of weighted lattice paths which, when summed over, give the stationary state probability distribution. We show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions.