The inactive-active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton

TL;DR

This paper studies the phase transition in noisy additive cellular automata, using mean field and finite-size scaling methods, and finds it belongs to the directed percolation universality class, with implications for related automata.

Contribution

It characterizes the critical behavior of noisy additive cellular automata and links it to directed percolation universality, also comparing mean field predictions with empirical results.

Findings

Phase transition belongs to directed percolation class.

Mean field approximations are reasonably accurate away from criticality.

Noisy elementary CA 90 and 102 share the same critical behavior.

Abstract

We investigate the inactive-active phase transition in an array of additive (exclusive-or) cellular automata under noise. The model is closely related with the Domany-Kinzel probabilistic cellular automaton, for which there are rigorous as well as numerical estimates on the transition probabilities. Here we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram's enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.