|
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 \alpha and exit from
the right with probability \beta. 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 \tau is
essentially the generating function for permutation tableaux of shape
\lambda(\tau) 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 certain Markov chain on the
permutations in S_{n+1}. Applications of our results include some
monotonicity results for the PASEP, and enumerative results for
permutations.
This work is joint with Sylvie Corteel.
|
