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Macdonald–Koornwinder moments and the two-species exclusion process

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Abstract

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 \(q=t\) are closely connected to the partition function for the two-species exclusion process.

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Authors and Affiliations

  1. Institut de Recherche en Informatique Fondamentale, CNRS et Universite Paris Diderot, Case 7014, 75205, Paris Cedex 13, France

    Sylvie Corteel

  2. Department of Mathematics, University of California, Berkeley, Evans Hall Room 913, Berkeley, CA, 94720, USA

    Lauren K. Williams

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  1. Sylvie Corteel
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  2. Lauren K. Williams
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Correspondence to Lauren K. Williams.

Additional information

SC was partially funded by the “Combinatoire à Paris” Projet Emergences 2013–2017 and by “ALEA Sorbonne” Projet IDEX USPC. LW was partially supported by the Fondation Sciences Mathématiques de Paris, the Simons foundation, a Rose-Hills Investigator award, and an NSF CAREER award. Both authors are grateful for the comments of the anonymous referees, and the support of the France-Berkeley fund, and would like to acknowledge the hospitality of LIAFA, where part of this work was carried out.

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Corteel, S., Williams, L.K. Macdonald–Koornwinder moments and the two-species exclusion process. Sel. Math. New Ser. 24, 2275–2317 (2018). https://doi.org/10.1007/s00029-017-0375-x

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