Macdonald–Koornwinder moments and the two-species exclusion process

• Sylvie Corteel ^[1] ; Lauren K. Williams ^[2]
  1. [1] CNRS et Universite Paris Diderot
  2. [2] University of California
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 3, 2018, págs. 2275-2317
• Idioma: inglés
• Enlaces
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• Resumen

  • 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.