Percolation and disorder-resistance in cellular automata

TL;DR

This paper proves that certain one-dimensional cellular automata exhibit disorder-resistance, leading to predictable periodic patterns from most initial seeds, while also allowing complex or chaotic behaviors from specific seeds, using a novel percolation approach.

Contribution

It introduces a new percolation-based method to analyze disorder-resistance and complex behaviors in cellular automata, including boundary dynamics of solidification rules.

Findings

Most initial seeds lead to periodic, predictable patterns.

Some seeds produce complex or chaotic evolution.

A new percolation theory variant is developed for analysis.

Abstract

We rigorously prove a form of disorder-resistance for a class of one-dimensional cellular automaton rules, including some that arise as boundary dynamics of two-dimensional solidification rules. Specifically, when started from a random initial seed on an interval of length $L$, with probability tending to one as $L\to\infty$, the evolution is a replicator. That is, a region of space-time of density one is filled with a spatially and temporally periodic pattern, punctuated by a finite set of other finite patterns repeated at a fractal set of locations. On the other hand, the same rules exhibit provably more complex evolution from some seeds, while from other seeds their behavior is apparently chaotic. A principal tool is a new variant of percolation theory, in the context of additive cellular automata from random initial states.