Tableaux combinatorics of the two-species PASEP

TL;DR

This paper introduces a combinatorial framework called rhombic alternative tableaux to compute stationary probabilities in a two-species PASEP with heavy and light particles on a one-dimensional lattice.

Contribution

It generalizes previous combinatorial results for PASEP by defining new tableaux for the two-species case, providing explicit formulas for stationary probabilities.

Findings

Defined rhombic alternative tableaux for two-species PASEP

Derived combinatorial formulas for stationary probabilities

Extended previous PASEP combinatorial models

Abstract

We study a two-species PASEP, in which there are two types of particles, "heavy" and "light," hopping right and left on a one-dimensional lattice of $n$ cells with open boundaries. In this process, only the "heavy" particles can enter on the left of the lattice and exit from the right of the lattice. In the bulk, any transition where a heavier particle type swaps places with an adjacent lighter particle type is possible. We generalize combinatorial results of Corteel and Williams for the ordinary PASEP by defining a combinatorial object which we call a rhombic alternative tableau that gives a combinatorial formula for the stationary probabilities for the states of this two-species PASEP.