A sharper threshold for bootstrap percolation in two dimensions

• 成果类型: Article
• 署名作者: Gravner, Janko; Holroyd, Alexander E.; Morris, Robert
• 署名单位: Instituto Nacional de Matematica Pura e Aplicada (IMPA); University of California System; University of California Davis; Microsoft; University of British Columbia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0338-z
• 发表日期: 2012
• 页码: 1-23
• 关键词: metastability threshold; GROWTH; models; nucleation; BEHAVIOR; trees

摘要

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.