Macdonald-Koornwinder moments and the two-species exclusion process.

Introduced in the late 1960’s (Macdonald et al. in Biopolymers 6:1-25, 1968; Spitzer in Adv Math 5:246-290, 1970), the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey-Wilson polynomials (Uchiyama et al. in J Phys A 37(18):4985-5002, 2004; Corteel and Williams in Duke Math J 159(3):385-415, 2011; Corteel et al. in Trans Am Math Soc 364(11):6009-6037, 2012), a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey-Wilson polynomials can be viewed as a specialization of the multivariate Macdonald-Koornwinder polynomials (also known as Koornwinder polynomials), which in turn give rise to the Macdonald polynomials associated to any classical root system via a limit or specialization (van Diejen in Compos Math 95(2):183-233, 1995). In light of the fact that Koornwinder polynomials generalize the Askey-Wilson polynomials, it is natural to ask whether one can find a particle model whose partition function is related to Koornwinder polynomials. In this article we answer this question affirmatively, by showing that Koornwinder moments at q=t<inline-graphic></inline-graphic> are closely connected to the partition function for the two-species exclusion process. [ABSTRACT FROM AUTHOR]