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 $$t>0$$ 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.

Duality for a boundary driven asymmetric model of energy transport

“…More recent results include [2] where a duality relation between an half-line open ASEP and a sub-Markov process where particles perform an asymmetric exclusion dynamics in the bulk and are killed at the boundary is proven. In [28,29] it is shown a reverse duality relation for an open ASEP with open boundary and a shock ASEP with reflecting boundary.…”

Recent advances in collective phenomena

“…The latter are discussed in the case of the Ising and Riviera models. Another paradigmatic and exactly solvable undimensional model, the asymmetric simple exclusion process (ASEP) model, is investigated in the fourth paper, "Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries" by Schütz [9]. In this model, particles are injected into the system at its boundaries.…”