Macdonald-Koornwinder moments and the two-species exclusion process

TL;DR

This paper establishes a connection between Koornwinder polynomial moments and the partition function of a two-species exclusion process, extending known links from Askey-Wilson polynomials to a broader class of orthogonal polynomials.

Contribution

It demonstrates that Koornwinder moments at q=t recover the two-species exclusion process partition function, generalizing previous results involving Askey-Wilson polynomials.

Findings

Koornwinder moments at q=t relate to the two-species exclusion process partition function.

Provides a hook length formula for Koornwinder moments at q=t=1.

Extends the connection between orthogonal polynomials and particle models to Koornwinder polynomials.

Abstract

Introduced in the late 1960's, 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, 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. 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 the "homogeneous" Koornwinder moments at q=t recover the partition function for the two-species exclusion process. We also provide a "hook length" formula for Koornwinder moments when q=t=1.