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.