Unpredictable paths and percolation

• 成果类型: Article
• 署名作者: Benjamini, I; Pemantle, R; Peres, Y
• 署名单位: Weizmann Institute of Science; University of Wisconsin System; University of Wisconsin Madison; Hebrew University of Jerusalem; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 1198-1211
• 关键词: supercritical bernoulli percolation; random-walks; cluster

摘要

We construct a nearest-neighbor process {S-n} on Z that is less predictable than simple random walk, in the sense that given the process until time n, the conditional probability that Sn+k = x is uniformly bounded by Ck(-alpha) for some alpha > 1/2. From this process, we obtain a probability measure mu on oriented paths in Z(3) such that the number of intersections of two paths, chosen independently according to mu, has an exponential tail. (For d greater than or equal to 4, the uniform measure on oriented paths from the origin in Z(d) has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter p is close enough to 1. This yields an extension of a theorem of Grimmett, Kesten and Zhang, who proved that supercritical percolation clusters in Z(d) are transient for all d greater than or equal to 3.