Indistinguishability of percolation clusters

• 成果类型: Article
• 署名作者: Lyons, R; Schramm, O
• 署名单位: Indiana University System; Indiana University Bloomington; Weizmann Institute of Science
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022874816
• 发表日期: 1999
• 页码: 1809-1836
• 关键词: infinite clusters; random-walks; property-t; uniqueness; models; graphs; trees

摘要

We show that when percolation produces infinitely many infinite dusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for p(u).