Exact results for one dimensional stochastic cellular automata for different types of updates

TL;DR

This paper derives exact stationary states for one-dimensional stochastic cellular automata with various update rules, revealing connections between different update types and providing explicit relations for densities and correlations.

Contribution

It introduces a matrix product formalism approach to construct stationary states and establishes a general link between states from different update schemes in stochastic cellular automata.

Findings

Explicit stationary states for different update types

Connections between densities and correlations across update schemes

General relations derived for stationary state properties

Abstract

We study two common types of time-noncontinuous updates for one dimensional stochastic cellular automata with arbitrary nearest neighbor interactions and arbitrary open boundary conditions. We first construct the stationary states using the matrix product formalism. This construction then allows to prove a general connection between the stationary states which are produced by the two different types of updates. Using this connection, we derive explicit relations between the densities and correlation functions for these different stationary states.