Combinatorics of a disordered two-species ASEP on a torus

TL;DR

This paper introduces a disordered two-species ASEP on a torus, providing a combinatorial understanding of its stationary distribution, and reveals a novel Scott Russell phenomenon relating particle currents.

Contribution

It defines a new disordered two-species ASEP on a torus, computes its stationary distribution, and explains its relation to the disordered ASEP on a ring with a novel current phenomenon.

Findings

Stationary weights are proportional to monomials in hopping rates.

Partition function, densities, and currents are explicitly computed.

Discovered the Scott Russell phenomenon linking horizontal and vertical currents.

Abstract

We define a new disordered asymmetric simple exclusion process (ASEP) with two species of particles, first-class particles labelled $\bullet$ and second-class particles labelled ${\scriptstyle \Box}$, on a two-dimensional toroidal lattice. The dynamics is controlled by particles labelled $\bullet$, which only move horizontally, with forward and backward hopping rates $p_i$ and $q_i$ respectively if the $\bullet$ is on row $i$. The motion of particles labelled ${\scriptstyle \Box}$ depends on the relative position of these with respect to $\bullet$'s, and can be both horizontal and vertical. We show that the stationary weight of any configuration is proportional to a monomial in the $p_i$'s and $q_i$'s. Our process projects to the disordered ASEP on a ring, and so explains combinatorially the stationary distribution of the latter first derived by Evans (Europhysics Letters, 1996). We compute the partition function, as well as densities and currents of $\bullet$'s and ${\scriptstyle \Box}$'s in the stationary state. We observe a novel mechanism we call the Scott Russell phenomenon: the current of ${\scriptstyle \Box}$'s in the vertical direction is the same as that of $\bullet$'s in the horizontal direction.