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 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 and , and they may exit and enter at the right with probabilities and . In the bulk, the probability of hopping left is 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 . 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.
"Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials." Duke Math. J. 159 (3) 385 - 415, 15 September 2011. https://doi.org/10.1215/00127094-1433385