Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson 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 of n sites with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters gamma=delta=0. Using our first result and also results of Uchiyama-Sasamoto-Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.

An announcement of these results appeared here: http://www.pnas.org/content/early/2010/03/25/0909915107.abstract This version of the paper has updated references and corrects a gap in the proof of Proposition 6.11 which was in the published version

References in corpus (3)

• Total positivity, Grassmannians, and networks
• Asymptotics in ASEP with Step Initial Condition
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries

Cited by in corpus (19)

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• Expansions of one density via polynomials orthogonal with respect to the other
• Combinatorics of the two-species ASEP and Koornwinder moments
• Open ASEP in the Weakly Asymmetric Regime
• Befriending Askey-Wilson polynomials
• Occupied corners in tree-like tableaux
• Nevanlinna Theory of the Wilson Divided-difference Operator
• Combinatorics of the asymmetric exclusion process on a semi-infinite lattice
• On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications
• Weighted random staircase tableaux
• Combinatorics of a disordered two-species ASEP on a torus
• Boundary lines and Askey-Wilson type moments
• The spectrum of an asymmetric annihilation process
• Askey--Wilson Integral and its Generalizations
• On the distribution of parameters in random weighted staircase tableaux
• Greek letters in random staircase tableaux
• On the norm of the $q$-circular operator