Universality for monotone cellular automata

TL;DR

This paper introduces a general method to analyze the growth of infected regions in monotone cellular automata, providing sharp bounds on percolation thresholds and supporting the Universality Conjecture across models.

Contribution

Develops a universal bounding technique for infected set growth in monotone cellular automata, confirming the Universality Conjecture for critical models.

Findings

Established a lower bound on the critical probability for percolation.

Proved bounds are sharp up to a constant factor in the exponent.

Supported the Universality Conjecture through rigorous analysis.

Abstract

In this paper we study monotone cellular automata in $d$ dimensions. We develop a general method for bounding the growth of the infected set when the initial configuration is chosen randomly, and then use this method to prove a lower bound on the critical probability for percolation that is sharp up to a constant factor in the exponent for every 'critical' model. This is one of three papers that together confirm the Universality Conjecture of Bollob\'as, Duminil-Copin, Morris and Smith.