Exact matrix-product states for parallel dynamics: Open boundaries and excess mass on the ring

TL;DR

This paper derives exact steady-state representations for ASEP models with open boundaries and ring geometries, revealing phase transitions and defect dynamics relevant for understanding shock formation in parallel update systems.

Contribution

It introduces a novel exact matrix-product state formulation combining scalar pair-factorized and matrix-product states for ASEP with open boundaries and ring geometries.

Findings

Exact steady-state weights expressed as a product of scalar and matrix states.

Identification of a phase transition between different defect velocities.

Exact calculation of defect velocities from the process-generating function.

Abstract

In this paper it is shown that the steady-state weights of the asymmetric simple exclusion process (ASEP) with open boundaries and parallel update can be written as a product of a scalar pair-factorized and a matrix-product state. This type of state is also obtained for an ASEP on a ring in which particles can move one or two sites. The dynamics leads to the formation of an excess hole that plays the role of a defect. We expect the process to play a similar role for parallel dynamics as the well-known ASEP with a single defect-particle (that is obtained in the continuous-time limit) especially for the study of shocks. The process exhibits a first-order phase transition between two phases with different defect velocities. These are calculated exactly from the process-generating function.