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15 September 2011 Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials
Sylvie Corteel, Lauren K. Williams
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Duke Math. J. 159(3): 385-415 (15 September 2011). DOI: 10.1215/00127094-1433385


Introduced in the late 1960s, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics that describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. 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 α and γ, and they may exit and enter at the right with probabilities β and δ. 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 γ=δ=0. Using our first result and also results of Uchiyama, Sasamoto, and Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980s 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.


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Sylvie Corteel. Lauren K. Williams. "Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials." Duke Math. J. 159 (3) 385 - 415, 15 September 2011.


Published: 15 September 2011
First available in Project Euclid: 29 August 2011

zbMATH: 1246.05170
MathSciNet: MR2831874
Digital Object Identifier: 10.1215/00127094-1433385

Primary: 05E10
Secondary: 60C05 , 82B23

Rights: Copyright © 2011 Duke University Press


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Vol.159 • No. 3 • 15 September 2011
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