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. In this talk we will
describe a surprising connection between the PASEP model and the
combinatorics of certain 0-1 tableaux called permutation tableaux. Namely,
we prove that in the long time limit, the probability that the PASEP model
is in a particular configuration tau is essentially the weight generating
function for permutation tableaux of shape lambda(tau).
This result suggests a connection between the PASEP model and total
positivity for the Grassmannian (via work of Postnikov). Additionally,
there should be connections to Stanley's theory of differential posets.