From multiline queues to Macdonald polynomials via the exclusion process
Sylvie Corteel, Olya Mandelshtam, Lauren Williams
Abstract
Recently James Martin introduced multiline queues, and used them to give a\ncombinatorial formula for the stationary distribution of the multispecies\nasymmetric simple exclusion exclusion process (ASEP) on a circle. The ASEP is a\nmodel of particles hopping on a one-dimensional lattice, which was introduced\naround 1970, and has been extensively studied in statistical mechanics,\nprobability, and combinatorics. In this article we give an independent proof of\nMartin's result, and we show that by introducing additional statistics on\nmultiline queues, we can use them to give a new combinatorial formula for both\nthe symmetric Macdonald polynomials P_{lambda}(x; q, t), and the nonsymmetric\nMacdonald polynomials E_{lambda}(x; q, t), where lambda is a partition. This\nformula is rather different from others that have appeared in the literature,\nsuch as the formulas due to Haglund, Haiman, and Loehr, the formula due to Ram\nand Yip, and the one due to Lenart. Our proof uses results of Cantini, de Gier,\nand Wheeler, who recently linked the multispecies ASEP on a circle to Macdonald\npolynomials.\n