We investigate the totally asymmetric exclusion process on Z, with the jump rate at site i given by ri = 1 for i 6= 0, r0 = r. It is easy to see that the maximal stationary current j(r) is nondecreasing in r and that j(r) = 1/4 for r ≥ 1; it is a long outstanding problem to determine whether or not the critical value rc of r such that j(r) = 1/4 for r > rc is strictly less than 1. Here we present a heuristic argument, based on the analysis of the first sixteen terms in a formal power series expansion of j(r) obtained from finite volume systems, that rc = 1 and that for r / 1, j(r) ≃ 1/4 − γ exp[−a/(1 − r)] with a ≈ 2. We also give some new exact results about this system; in particular we prove that j(r) = Jmax(r), with Jmax(r) the hydrodynamic maximal current defined by Sepp¨al¨ainen, and thus establish continuity of j(r). Finally we describe a related exactly solvable model, a semi-infinite system in which the site i = 0 is always occupied. For that system, r s-i c = 1/2 and the analogue j s-i(r) of j(r) satisfies j s-i(r) = r(1 − r) for r ≤ r s-i c ; j s-i(r) is the limit of finite volume currents inside the curve |r(1 − r)| = 1/4 in the complex r plane and we suggest that analogous behavior may hold for the original system.
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