PLANAR LATTICES DO NOT RECOVER FROM FOREST FIRES

• 成果类型: Article
• 署名作者: Kiss, Demeter; Manolescu, Ioan; Sidoravicius, Vladas
• 署名单位: University of Cambridge; University of Geneva; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP958
• 发表日期: 2015
• 页码: 3216-3238
• 关键词: self-destructive percolation

摘要

Self-destructive percolation with parameters p, delta is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to infinite clusters, then opening every closed site with probability delta, independently of the rest. Call theta (p, delta) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two-dimensional lattices, we show the existence of delta > 0 such that, for any p > p(c), theta (p, delta) = 0. This proves the conjecture of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480-501], who introduced the model. Our results combined with those of van den Berg and Brouwer [Random Structures Algorithms 24 (2004) 480-501] imply the nonexistence of the infinite parameter forest-fire model. The methods herein apply to site and bond percolation on any two-dimensional planar lattice with sufficient symmetry.