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. In this paper we prove a close connection between the PASEP and the combinatories 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 inatrix 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 occurrences of generalized patterns. (c) 2006 Elsevier Inc. All rights reserved.