From multiline queues to Macdonald polynomials via the exclusion process

Abstract

Recently James Martin introduced multiline queues, and used them to give a combinatorial formula for the stationary distribution of the multispecies asymmetric simple exclusion exclusion process (ASEP) on a circle. The ASEP is a model of particles hopping on a one-dimensional lattice, which was introduced around 1970, and has been extensively studied in statistical mechanics, probability, and combinatorics. In this article we give an independent proof of Martin s result, and we show that by introducing additional statistics on multiline queues, we can use them to give a new combinatorial formula for both the symmetric Macdonald polynomials P_{lambda}(x; q, t), and the nonsymmetric Macdonald polynomials E_{lambda}(x; q, t), where lambda is a partition. This formula is rather different from others that have appeared in the literature, such as the Haglund-Haiman-Loehr formula. Our proof uses results of Cantini-de Gier-Wheeler, who recently linked the multispecies ASEP on a circle to Macdonald polynomials.