Critical percolation on any nonamenable group has no infinite clusters

• 成果类型: Article
• 署名作者: Benjamini, I; Lyons, R; Peres, Y; Schramm, O
• 署名单位: Weizmann Institute of Science; Hebrew University of Jerusalem; University of California System; University of California Berkeley; Indiana University System; Indiana University Bloomington
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1999
• 页码: 1347-1356
• 关键词: uniqueness; trees

摘要

We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasi-transitive graphs with a unimodular automorphism group. The key tool is a mass-transport method, which is a technique of averaging in nonamenable settings.