Multi-Catalan Tableaux and the Two-Species TASEP

TL;DR

This paper provides a combinatorial formula for the steady state probabilities of a two-species PASEP model using multi-Catalan tableaux, with explicit results at q=0 and conjectures for q=1.

Contribution

It introduces a novel combinatorial interpretation of the two-species PASEP's stationary distribution at q=0 using multi-Catalan tableaux and proposes a conjecture extending to q=1.

Findings

Explicit determinantal formula for q=0 stationary probabilities

A Markov process on tableaux projecting to the PASEP

Conjecture for q=1 stationary distribution using two-species alternative tableaux

Abstract

The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the two-species PASEP. In this model, there are two species of particles, one "heavy" and one "light", on a one-dimensional finite lattice with open boundaries. Both particles can swap places with adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can swap places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our main result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and thus directly explains the connection between the two. Finally, we give a conjecture that extends our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableau.