Duke Mathematical Journal

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Abstract

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 $\alpha$ 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, 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.

Article information

Source
Duke Math. J., Volume 159, Number 3 (2011), 385-415.

Dates
First available in Project Euclid: 29 August 2011

https://projecteuclid.org/euclid.dmj/1314648275

Digital Object Identifier
doi:10.1215/00127094-1433385

Mathematical Reviews number (MathSciNet)
MR2831874

Zentralblatt MATH identifier
1246.05170

Citation

Corteel, Sylvie; Williams, Lauren K. Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials. Duke Math. J. 159 (2011), no. 3, 385--415. doi:10.1215/00127094-1433385. https://projecteuclid.org/euclid.dmj/1314648275

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