The survival probability for critical spread-out oriented percolation above 4+1 dimensions. I. Induction

• 成果类型: Article
• 署名作者: van der Hofstad, Remco; den Hollander, Frank; Slade, Gordon
• 署名单位: Leiden University; Leiden University - Excl LUMC; Eindhoven University of Technology; University of British Columbia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-006-0028-z
• 发表日期: 2007
• 页码: 363-389
• 关键词: incipient infinite cluster; contact process

摘要

We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/Bn-2 as n -> infinity, where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/Bn-2 as n -> infinity. The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction.