Two applications of percolation to cellular automata

Abstract

The point of this paper is to show how ideas from percolation can be used to study the asymptotic behavior of some cellular automata systems. In particular, using these ideas, we prove that the Greenberg-Hastings and cyclic cellular automata models with three colors, threshold 2, and theL ∞ neighborhood are uniformly asymptotically locally periodic ind≥2 dimensions. We also show that every lattice point is eventually “controlled by a finite clock” in the standard Greenberg-Hastings and cyclic cellular automata models in two dimensions, which is a stronger description than the already known asymptotic behavior.