Tableaux combinatorics for the asymmetric exclusion process

The partially asymmetric exclusion process (PASEP) 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. It is partially asymmetric in the sense that the probability of hopping left is $q$ times the probability of hopping right. Additionally, particles may enter from the left with probability $\alpha$ and exit from the right with probability $\beta$. In this paper we prove a close connection between the PASEP and the combinatorics of permutation tableaux. (These tableaux come indirectly from the totally nonnegative part of the Grassmannian, via work of Postnikov, and were studied in a paper of Steingrimsson and the second author.) Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration $\tau$ is essentially the generating function for permutation tableaux of shape $\lambda(\tau)$ enumerated according to three statistics. The proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on the {\it matrix ansatz} for the PASEP model. As an application, we prove some monotonicity results for the PASEP. We also derive some enumerative consequences for permutations enumerated according to various statistics such as weak excedence set, descent set, crossings, and occurences of generalized patterns.Comment: Clarified exposition, more general result, new author (SC), 19 pages, 6 figure

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

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.Comment: 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 versio

The Asymptotic Distribution of Symbols on Diagonals of Random Weighted Staircase Tableaux

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, the distri- bution of parameters on the first diagonal was proven to be asymptotically normal. In that same paper, a conjecture was made that the other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for fixed k. In particular, we prove that the distribution of the number of alphas (betas) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1\2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1

A Determinantal Formula for Catalan Tableaux and TASEP Probabilities

We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with $\alpha$'s and $\beta$'s that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with $n$ sites, precisely $k$ of the sites are occupied by particles. This formula is an $\alpha\ /\ \beta$ generalization of the Narayana numbers.Comment: 19 pages, 12 figure