A sharper threshold for bootstrap percolation in two dimensions

TL;DR

This paper refines the understanding of the critical probability in two-dimensional bootstrap percolation, providing a more precise asymptotic expansion that corrects previous numerical predictions.

Contribution

It improves the asymptotic estimate of the critical probability by determining the second term in the expansion up to a poly(log log n)-factor.

Findings

Sharpens the asymptotic expansion of p_c with a second term of -(log n)^{-3/2+ o(1)}.

Corrects numerical predictions from physics literature.

Provides a more precise estimate of the critical probability as n grows large.

Abstract

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim \pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical predictions from the physics literature.