Séminaire Lotharingien de Combinatoire, 80B.95 (2018), 12 pp.
Olya Mandelshtam
Bijection from Multiline Queues to Rhombic Tableaux for the Inhomogeneous 2-TASEP
Abstract.
The 2-TASEP is a model describing the dynamics of first and second
class particles hopping in one direction on a finite 1D lattice. For
the 2-TASEP with periodic boundary conditions, there is a well-known
description for the stationary probabilities in terms of
multiline queues of Ferarri and Martin. On the other hand, for
the 2-TASEP with open boundary conditions, there is a rich connection
to tableaux combinatorics: its stationary probabilities are described
using rhombic alternative tableaux. In this article, we unify
the two approaches by defining a new object, the toric rhombic
tableaux and describing a simple bijection between these tableaux and
multiline queues for the 2-TASEP with periodic boundary
conditions. Furthermore, with a natural modification of both the
rhombic alternative tableaux and the toric rhombic tableaux, we obtain
a tableaux interpretation for probabilities of the inhomogeneous
2-TASEP both with periodic and open boundary conditions, in which
different classes of particles hop with different rates. Through our
bijection, our result generalizes a result of Ayyer and Linusson on
multiline queues.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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