Bijection from Multiline Queues to Rhombic Tableaux for the Inhomogeneous 2-TASEP

Olya Mandelshtam

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

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.