Universality for two-dimensional critical cellular automata

TL;DR

This paper establishes a universal threshold order for percolation in all critical two-dimensional bootstrap percolation models, unifying and extending previous results in the field.

Contribution

It determines the percolation threshold order for all critical bootstrap percolation models in two dimensions, providing a comprehensive universality result.

Findings

Threshold order for critical models is now fully characterized.

Includes and extends previous specific case results.

Strengthens bounds on percolation thresholds.

Abstract

We study the class of monotone, two-state, deterministic cellular automata, in which sites are activated (or 'infected') by certain configurations of nearby infected sites. These models have close connections to statistical physics, and several specific examples have been extensively studied in recent years by both mathematicians and physicists. This general setting was first studied only recently, however, by Bollob\'as, Smith and Uzzell, who showed that the family of all such 'bootstrap percolation' models on $\mathbb{Z}^2$ can be naturally partitioned into three classes, which they termed subcritical, critical and supercritical. In this paper we determine the order of the threshold for percolation (complete occupation) for every critical bootstrap percolation model in two dimensions. This 'universality' theorem includes as special cases results of Aizenman and Lebowitz, Gravner and Griffeath, Mountford, and van Enter and Hulshof, significantly strengthens bounds of Bollob\'as, Smith and Uzzell, and complements recent work of Balister, Bollob\'as, Przykucki and Smith on subcritical models.