Phase transitions in random mixtures of elementary cellular automata

TL;DR

This paper explores phase transitions in a class of probabilistic cellular automata formed by mixing elementary rules, using analytical methods to identify conditions under which phase transitions occur.

Contribution

It introduces a comprehensive analytical framework to study phase transitions in mixed elementary cellular automata without relying on local properties.

Findings

Identifies phase transition conditions for all mixtures with the null rule.

Uses Mean Field and Dobrushin Criterion for analysis.

Results align with previous numerical and rigorous studies.

Abstract

We investigate one-dimensional Probabilistic Cellular Automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixture of two different Elementary Cellular Automata rules. All the cells are updated synchronously and the probability for one cell to be $0$ or $1$ at time $t$ depends only on the value of the same cell and that of its neighbors at time $t-1$. These very simple models show a very rich behavior strongly depending on the choice of the two Elementary Cellular Automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates $0$ to any possible local configuration, with any of the other $255$ elementary rule. We approach the problem analytically via a Mean Field approximation and via the use of a rigorous approach based on the application of the Dobrushin Criterion. The distinguishing trait of our result is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are coherent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.