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 • and second-class particles labelled □, on a two-dimensional toroidal lattice. The dynamics is controlled by particles labelled •, which only move horizontally, with forward and backward hopping rates p_i and q_i respectively if the • is on row i. The motion of particles labelled □ depends on the relative position of these with respect to •’s, and can be both horizontal and vertical. We show that the stationary weight of any configuration is proportional to a monomial in the ’s and ’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 •’s and □’s in the stationary state. We observe a novel mechanism we call the Scott Russell phenomenon: the current of □’s in the vertical direction is the same as that of •’s in the horizontal direction.