Tableaux combinatorics for the asymmetric exclusion process

Details Tableaux combinatorics for the asymmetric exclusion process Tableaux combinatorics for the asymmetric exclusion process

2006-02-07

arxiv.org/abs/math/0602109v3

Mathematics

Combinatorics

Clarified exposition, more general result, new author (SC), 19 pages, 6 figures

Scientific paper

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 $\beta$. In this paper we prove 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, and were studied in a paper of Steingrimsson and the second author.) Namely, we prove that 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. The proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on the {\it matrix ansatz} for the PASEP model. As an application, we prove some monotonicity results for the PASEP. We also derive some enumerative consequences for permutations enumerated according to various statistics such as weak excedence set, descent set, crossings, and occurences of generalized patterns.

Affiliated with

Also associated with

No associations

Rating

Tableaux combinatorics for the asymmetric exclusion process does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Tableaux combinatorics for the asymmetric exclusion process, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Tableaux combinatorics for the asymmetric exclusion process will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-63375