  We study two versions of the asymmetric exclusion process (ASEP) -- an ASEP
on a semi-infinite lattice 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.
