Infinite clusters in dependent automorphism invariant percolation on trees

• 成果类型: Article
• 署名作者: Haggstrom, O
• 署名单位: Chalmers University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1423-1436
• 关键词: random-walks

摘要

We study dependent bond percolation on the homogeneous tree T-n of order n greater than or equal to 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clusters a.s. is either 0 or infinity. Furthermore, each infinite cluster a.s. has either 1, 2 or infinitely many topological ends, and infinite clusters with infinitely many topological ends have a.s. a branching,Umber greater than 1. We also show that if the marginal probability that a single edge is open is at least 2/(n + 1), then the existence of infinite clusters has to have positive probability. Several concrete examples are considered.