### 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.