ar X iv : m at h / 06 02 10 9 v 2 [ m at h . C O ] 2 1 A pr 2 00 6 PERMUTATION TABLEAUX AND THE ASYMMETRIC EXCLUSION PROCESS

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a aystem 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 α and exit from the right with probability β. 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 [12], and were studied in a paper of Steingrimsson and the second author [15].) Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration τ is essentially the generating function for permutation tableaux of shape λ(τ) enumerated according to three statistics. The proof of this result uses a result of Derrida et al [7] on the matrix ansatz for the PASEP model. As an application, we prove some monotonicity results for the PASEP when α = β = 1. 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.