Increasing coupling of probabilistic cellular automata

TL;DR

This paper establishes a necessary and sufficient condition for creating increasing couplings of multiple probabilistic cellular automata, with applications to attractive systems and insights into limitations when the state space is only partially ordered.

Contribution

It provides a comprehensive construction theorem for increasing couplings in totally ordered spaces and extends some results to certain partially ordered spaces.

Findings

A necessary and sufficient condition for increasing coupling in totally ordered spaces.

Examples showing non-existence of couplings for more than two dynamics in partially ordered spaces.

Extension of main results to specific classes of partially ordered spaces.

Abstract

We give a necessary and sufficient condition for the existence of an increasing coupling of $N$ ($N \geq 2$) synchronous dynamics on $S^{\mathbb Z^d}$(PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where $S$ is totally ordered, applications to attractive PCA's are given. When $S$ is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.