Duality relations for the periodic ASEP conditioned on a low current

Abstract

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of currents, parametrized by the driving strength sK, K ≥ 1, we prove duality relations which arise from the quantum algebra Uq[gl(2)] symmetry of the generator of the process with reflecting boundary conditions. Using these duality relations we prove on microscopic level a travelling-wave property of the conditioned process for a family of shock-antishock measures for N > K particles: If the initial measure is a member of this family with K microscopic shocks at positions (x1, …, xK), then the measure at any time t > 0 of the process with driving strength sK is a convex combination of such measures with shocks at positions (y1, …, yK), which can be expressed in terms of K-particle transition probabilities of the conditioned ASEP with driving strength sN.