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

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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Published 11 March 2015