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 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 semiinfinite ASEP was first studied by Liggett [6] and then Grosskinsky [5], while the finite ASEP had been introduced earlier by Spitzer [8] and MacdonaldGibbs-Pipkin [7]. 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 β and δ, respectively, and we set δ = −β, 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 [2, 3], we obtain a combinatorial formula for finite correlation functions of the ASEP on a semi-infinite lattice.