Exact density profiles for the fully asymmetric exclusion process with discrete-time dynamics on semi-infinite chains

Exact density profiles in the steady state of the one-dimensional fully asymmetric simple-exclusion process on a semi-infinite chain are obtained in the case of forward-ordered sequential dynamics by taking the thermodynamic limit in our recent exact results for a finite chain with open boundaries. The corresponding results for sublattice-parallel dynamics follow from the relationship obtained by Rajewsky and Schreckenberg [Physica A 245, 139 (1997)], and for parallel dynamics from the mapping found by Evans, Rajewsky, and Speer [J. Stat. Phys. 95, 45 (1999)]. Our analytical expressions involve Laplace-type integrals, rather than complicated combinatorial expressions, which makes them convenient for taking the limit of a semi-infinite chain, and for deriving the asymptotic behavior of the density profiles at large distances from its end. By comparing the asymptotic results appropriate for parallel update with those published in the above cited paper by Evans, Rajewsky, and Speer, we find complete agreement except in two cases, in which we correct technical errors in the final results given there.