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Annales de l’Institut Henri Poincaré D

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Volume 3, Issue 3, 2016, pp. 321–348
DOI: 10.4171/AIHPD/30

Published online: 2016-09-14

Multi-Catalan tableaux and the two-species TASEP

Olya Mandelshtam[1]

(1) University of California, Berkeley, USA

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.

Keywords: TASEP, multispecies, tableaux

Mandelshtam Olya: Multi-Catalan tableaux and the two-species TASEP. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3 (2016), 321-348. doi: 10.4171/AIHPD/30