Temporal evolution of product shock measures in TASEP with sublattice-parallel update

It is known that when the steady state of a one-dimensional multispecies system, which evolves via a random-sequential updating mechanism, is written in terms of a linear combination of Bernoulli shock measures with random-walk dynamics, it can be equivalently expressed as a matrix-product state. In this case the quadratic algebra of the system always has a two-dimensional matrix representation. Our investigations show that this equivalence exists at least for the systems with deterministic sublattice-parallel update. In this paper we consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries and sublattice-parallel update as an example.