Journal of Combinatorics

Volume 5 (2014)

Number 4

Combinatorics of the asymmetric exclusion process on a semi-infinite lattice

Pages: 419 – 434

DOI: http://dx.doi.org/10.4310/JOC.2014.v5.n4.a1

Authors

Tomohiro Sasamoto (Department of Mathematics, Chiba University, Chiba, Japan)

Lauren Williams (Department of Mathematics, University of California at Berkeley)

Abstract

We study two versions of the asymmetric exclusion process (ASEP)—an ASEP on a semi-infinite lattice $\mathbb{Z}^+$ with an open left boundary, and an ASEP on a finite lattice with open left and right boundaries—and we demonstrate a surprising relationship between their stationary measures. The semi-infinite ASEP was first studied by Liggett and then Grosskinsky, while the finite ASEP had been introduced earlier by Spitzer and Macdonald-Gibbs-Pipkin. We show that the finite correlation functions involving the first $L$ sites for the stationary measures on the semi-infinite ASEP can be obtained as a nonphysical specialization of the stationary distribution of an ASEP on a finite one-dimensional lattice with $L$ sites. Namely, if the output and input rates of particles at the right boundary of the finite ASEP are $\beta$ and $\delta$, respectively, and we set $\delta = -\beta$, then this specialization corresponds to sending the right boundary of the lattice to infinity. Combining this observation with work of the second author and Corteel, we obtain a combinatorial formula for finite correlation functions of the ASEP on a semi-infinite lattice.

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