The second term for two-neighbour bootstrap percolation in two dimensions

TL;DR

This paper refines the critical probability threshold for two-neighbour bootstrap percolation on a 2D grid, providing a more precise asymptotic formula and introducing technical innovations with broader applications.

Contribution

It improves the lower bound for the critical probability in 2D bootstrap percolation, offering a sharper asymptotic estimate and new methods for analyzing droplet growth.

Findings

Critical probability p_c([n]^2,2) = (π^2/18)/log n - Θ(1)/(log n)^{3/2}

Enhanced understanding of droplet growth dynamics in bootstrap percolation

Potential applications to cellular automata and higher-dimensional processes

Abstract

In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected (in each time step) if they have at least $r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising model of ferromagnetism, and kinetically constrained spin models of the liquid-glass transition, the most extensively-studied case is the two-neighbour bootstrap process on the two-dimensional grid $[n]^2$. Around 15 years ago, in a major breakthrough, Holroyd determined the sharp threshold for percolation in this model, and his bounds were subsequently sharpened further by Gravner and Holroyd, and by Gravner, Holroyd and Morris. In this paper we strengthen the lower bound of Gravner, Holroyd and Morris by proving that the critical probability $p_c\big( [n]^2,2 \big)$ for percolation in the two-neighbour model on $[n]^2$ satisfies \[p_c\big( [n]^2,2 \big) = \frac{\pi^2}{18\log n} - \frac{\Theta(1)}{(\log n)^{3/2}}\,.\] The proof of this result requires a very precise understanding of the typical growth of a critical droplet, and involves a number of technical innovations. We expect these to have other applications, for example, to the study of more general two-dimensional cellular automata, and to the $r$-neighbour process in higher dimensions.