Tableaux Combinatorics for the Asymmetric Exclusion Process II

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 has been cited as a model for traffic flow and protein synthesis. In its most general form, particles may enter and exit at the left with probabilities α and γ, and they may exit and enter at the right with probabilities β and δ. In the bulk, the probability of hopping left is q times the probability of hopping right. In previous work [4] we used the matrix ansatz to give a combinatorial formula for the steady state probability of each state of the PASEP, when γ = δ = 0. The formula was the generating function for permutation tableaux of a fixed shape, weighted according to three statistics. In this paper we give a simple one-parameter generalization of the matrix ansatz, then use it to generalize our results about the PASEP to the case of general α, β, γ, δ (and q = 1). We replace permutation tableaux by the slightly more general bordered permutation tableaux, which we show have cardinality 4nn!. We also state our results in terms of alternative tableaux.