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 neighbors. 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 ∼ π²/(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.