We study the asymmetric zero-range process (ZRP) with L sites and open
boundaries, conditioned to carry an atypical current. Using a generalized Doob
h-transform we compute explicitly the transition rates of an effective process
for which the conditioned dynamics are typical. This effective process is a
zero-range process with renormalized hopping rates, which are space dependent
even when the original rates are constant. This leads to non-trivial density
profiles in the steady state of the conditioned dynamics, and, under generic
conditions on the jump rates of the unconditioned ZRP, to an intriguing
supercritical bulk region where condensates can grow. These results provide a
microscopic perspective on macroscopic fluctuation theory (MFT) for the weakly
asymmetric case: It turns out that the predictions of MFT remain valid in the
non-rigorous limit of finite asymmetry. In addition, the microscopic results
yield the correct scaling factor for the asymmetry that MFT cannot predict.Comment: 26 pages, 4 figure
