Shocks and Antishocks in the ASEP Conditioned on a Low Current

Abstract

We study the time evolution of the ASEP conditioned on an atypically low current up to a finite time t on a one-dimensional lattice with L sites for both periodic and open boundary conditions. For periodic boundary conditions we prove that at a specific value of the conditioned global current there is a one-parameter family of shock initial measures such that the shock position performs a biased random walk and that the measure seen from the shock position remains invariant. The density profile seen from the shock position is a hyperbolic tangent. For open boundary conditions we obtain a similar result for antishocks, but at a different value of the conditioned current and with a density profile that is a step function. For both cases we compute explicitly the jump rates of the shock (antishock) random walk. We comment on the relation of our microscopic results to the predictions of the macroscopic large deviation theory for the ASEP.