Brandeis University Everytopic Seminar

Friday November 3, 2006, Goldsmith 226

Speaker: Lauren Williams (Harvard)

Title: Tableaux combinatorics for the asymmetric exclusion process

Abstract: 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 α and exit from the right with probability β. We will explain 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.) Namely, 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. One of our proofs of this result reveals a hidden structure behind the PASEP: namely, the PASEP can be viewed as a quotient of a Markov chain on the set of permutations on n +1 letters. Applications of our results include some monotonicity results for the PASEP, and enumerative results for permutations.

This work is joint with Sylvie Corteel.