Subcritical monotone cellular automata

TL;DR

This paper proves that for subcritical monotone cellular automata in any dimension, the probability of initial activation needed for percolation is strictly positive, confirming a long-standing conjecture.

Contribution

It extends the proof of non-zero critical probability from two dimensions to all higher dimensions for subcritical monotone cellular automata.

Findings

Critical probability is non-zero for all subcritical models in any dimension.

Confirms a conjecture previously proven only in two dimensions.

Advances understanding of phase transition behavior in cellular automata.

Abstract

We study monotone cellular automata (also known as $\mathcal{U}$-bootstrap percolation) in $\mathbb{Z}^d$ with random initial configurations. Confirming a conjecture of Balister, Bollob\'as, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non-zero for all subcritical models.