Weakly robust periodic solutions of one-dimensional cellular automata with random rules

TL;DR

This paper investigates the likelihood of weakly robust periodic solutions in large-state one-dimensional cellular automata with random rules, showing they are rare and quantifying their probability.

Contribution

It provides a probabilistic analysis of WRPS existence in cellular automata with many states, using random graph theory and Poisson approximation techniques.

Findings

Probability of WRPS existence is asymptotically proportional to 1/n.

WRPS are rare in large-state automata, especially under certain divisibility conditions.

The analysis employs advanced probabilistic tools like the Chen-Stein method.

Abstract

We study $2$-neighbor one-dimensional cellular automata with a large number $n$ of states and randomly selected rules. We focus on the rules with weakly robust periodic solutions (WRPS). WRPS are global configurations that exhibit spatial and temporal periodicity and advance into any environment with at least a fixed strictly positive velocity. Our main result quantifies how unlikely WRPS are: the probability of existence of a WRPS within a finite range of periods is asymptotically proportional to $1/n$, provided that a divisibility condition is satisfied. Our main tools come from random graph theory and the Chen-Stein method for Poisson approximation.