A sharper threshold for bootstrap percolation in two dimensions

Abstract

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbours. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × 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 ~ π 2/(18 log n) as n → ∞. 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. © 2010 The Author(s).