Combinatorics of a disordered two-species ASEP on a torus

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.