A sharper threshold for bootstrap percolation in two dimensions

作者：Gravner Janko; Holroyd Alexander E; Morris Robert^*

来源：Probability Theory and Related Fields, 2012, 153(1-2): 1-23.

DOI：10.1007/s00440-010-0338-z

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 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) similar to pi (2)/(18 log n) as n -> a. 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.

• 出版日期2012-6
• 单位Microsoft