Cylindric rhombic tableaux and the two-species ASEP on a ring

The asymmetric simple exclusion exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice of n sites. It was introduced around 1970, and since then has been extensively studied by researchers in statistical mechanics, probability, and combinatorics. Recently the ASEP on a lattice with open boundaries has been linked to Koornwinder polynomials, and the ASEP on a ring has been linked to Macdonald polynomials. In this article we study the two-species asymmetric simple exclusion process (ASEP) on a ring, in which two kinds of particles ("heavy" and "light"), as well as "holes," can hop both clockwise and counterclockwise (at rates 1 or t depending on the particle types) on a ring of n sites. We introduce some new tableaux on a cylinder called cylindric rhombic tableaux (CRT), and use them to give a formula for the stationary distribution of the two-species ASEP -- each probability is expressed as a sum over all CRT of a fixed type. When lambda is a partition in {0,1,2}^n, we then give a formula for the nonsymmetric Macdonald polynomial E_{lambda} and the symmetric Macdonald polynomial P_{lambda} by refining our tableaux formulas for the stationary distribution.

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