Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

• 成果类型: Article
• 署名作者: Häggström, O; Peres, Y
• 署名单位: Chalmers University of Technology; University of California System; University of California Berkeley; Hebrew University of Jerusalem
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050208
• 发表日期: 1999
• 页码: 273-285
• 关键词: probability; continuity; trees; MODEL

摘要

Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p(2) > p(1) > p(c), each infinite p(2)-cluster contains an infinite pr-cluster; this yields an extension of Alexander's (1995) simultaneous uniqueness theorem. As a corollary, we obtain that the probability theta(v) (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p > p(c). All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.