Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

• 成果类型: Article
• 署名作者: Hermon, Jonathan; Hutchcroft, Tom
• 署名单位: University of British Columbia; University of Cambridge
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-01011-3
• 发表日期: 2021
• 页码: 445-486
• 关键词: anchored expansion; random-walks; bernoulli-percolation; transitive graphs; critical-behavior; PHASE-TRANSITION; uniqueness; probabilities; inequalities; sharpness

摘要

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > p(c)(G) then there exists a positive constant c(p) such that P-p(n <= vertical bar K vertical bar < infinity) <= e(-cpn) for every n >= 1, where K is the cluster of the origin. We deduce the following two corollaries: 1. Every infinite cluster in supercritical percolation on a transitive non-amenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56-84, 1999). 2. For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.