A sharper threshold for bootstrap percolation in two dimensions

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.