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

TL;DR

This paper explores the relationship between shock measures and matrix-product states in the TASEP with sublattice-parallel update, extending known results from random-sequential updates to deterministic updates.

Contribution

It demonstrates that the equivalence between shock measures and matrix-product states holds for TASEP with sublattice-parallel update, showing a two-dimensional matrix representation of the quadratic algebra.

Findings

Shock measures exhibit random-walk dynamics under sublattice-parallel update.

Matrix-product state representation is valid for TASEP with deterministic sublattice-parallel update.

The quadratic algebra has a two-dimensional matrix representation in this setting.

Abstract

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.