RANDOM INTERLACEMENTS AND AMENABILITY

• 成果类型: Article
• 署名作者: Teixeira, Augusto; Tykesson, Johan
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS); Instituto Nacional de Matematica Pura e Aplicada (IMPA); Weizmann Institute of Science
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP860
• 发表日期: 2013
• 页码: 923-956
• 关键词: discrete cylinders; random-walk; infinite clusters; vacant set; percolation; uniqueness; nonuniqueness

摘要

We consider the model of random interlacements on transient graphs, which was first introduced by Sznitman [Ann. of Math. (2) (2010) 171 2039-2087] for the special case of Z(d) (with d >= 3). In Sznitman [Ann. of Math. (2) (2010) 171 2039-2087], it was shown that on Z(d): for any intensity u > 0, the interlacement set is almost surely connected. The main result of this paper says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In particular, we show that in nonamenable transitive graphs, for small values of the intensity u the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of u. Finally, we establish the monotonicity of the transition between the disconnected and the connected phases, providing the uniqueness of the critical value u(c) where this transition occurs.