Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries

Abstract

A reverse duality between the asymmetric simple exclusion process (ASEP) with open boundary conditions and a biased random walk of a single particle is proved for a special manifold of boundary parameters of the ASEP. The duality function is given by the configuration probabilities of a family of Bernoulli shock measures with a microscopic shock at site x of the lattice. The boundary conditions of the dual random walk, which can be reflecting or absorbing, depend on the choice of the duality function. As a consequence of this duality, the full time-dependent distribution of the open ASEP starting from a fixed shock measure at x is given for any time t>0 by a convex combination of shock measures with shock at position y. The coefficients are the transition probabilities P(x, t|y, 0) at time t of the random walk starting at site y. It is also shown that this family of shock measures arises from a similarity transformation of the two-dimensional representation of the matrix algebra for the stationary matrix product measure of the open ASEP. This implies a further duality between random walks with different boundary conditions.