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 formulas due to Haglund, Haiman, and Loehr, the formula due to Ram and Yip, and the one due to Lenart. Our proof uses results of Cantini, de Gier, and Wheeler, who recently linked the multispecies ASEP on a circle to Macdonald polynomials.