2018/11/02 by Corteel, Sylvie, Mandelshtam, Olya, Williams, Lauren
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.01047
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 0,1,2n, we then give a formula for the nonsymmetric Macdonald polynomial Elambda and the symmetric Macdonald polynomial Plambda by refining our tableaux formulas for the stationary distribution.