Multi-Catalan tableaux and the two-species TASEP

Mandelshtam, Olya

Annales de l’Institut Henri Poincaré D, Tome 3 (2016) no. 3, pp. 321-348.

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.

Classification : 05-XX
Mots-clés : TASEP, multispecies, tableaux