Abstract
The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the two-species ASEP. 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
particles 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 and the partition function, 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 ASEP, and thus directly explains the connection between the two. Finally, we give a conjecture that gives a formula for the stationary distribution to the
q = 1
case, using certain two-species alternative tableaux.