The blockage problem

We investigate the totally asymmetric exclusion process on Z, with the jump rate at site i given by r_i=1 for i nonzero, r_0=r. It is easy to see that the maximal stationary current j® is nondecreasing in r and that j®=1/4 for r>=1; it is a long outstanding problem to determine whether or not the critical value r_c of r such that j®=1/4 for r>r_c 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® obtained from finite volume systems, that r_c=1 and that for r less than 1 and near 1, j® behaves as 1/4-\gamma\exp[-{a/(1-r)}] with a approximately equal to 2. We also give some new exact results about this system; in particular we prove that j®=J_max®, with J_max® the hydrodynamic maximal current defined by Seppalainen, and thus establish continuity of j®. Finally we describe a related exactly solvable model, a semi-infinite system in which the site i=0 is always occupied. For that system, the critical r is 1/2 and the analogue j_s® of j® satisfies j_s®=r(1-r) for r<=1/2; j_s® 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.