Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution
Authors
Hermon, Jonathan
Hutchcroft, Tom
Publication Date
2020-10-22Journal Title
Inventiones mathematicae
ISSN
0020-9910
Publisher
Springer Berlin Heidelberg
Volume
224
Issue
2
Pages
445-486
Language
en
Type
Article
This Version
VoR
Metadata
Show full item recordCitation
Hermon, J., & Hutchcroft, T. (2020). Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution. Inventiones mathematicae, 224 (2), 445-486. https://doi.org/10.1007/s00222-020-01011-3
Description
Funder: University of Cambridge
Abstract
Abstract: 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>pc(G) then there exists a positive constant cp such that Pp(n≤|K|<∞)≤e-cpnfor every n≥1, where K is the cluster of the origin. We deduce the following two corollaries: Every infinite cluster in supercritical percolation on a transitive nonamenable 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). 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.
Keywords
Article
Identifiers
s00222-020-01011-3, 1011
External DOI: https://doi.org/10.1007/s00222-020-01011-3
This record's URL: https://www.repository.cam.ac.uk/handle/1810/329768
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
Statistics
Total file downloads (since January 2020). For more information on metrics see the
IRUS guide.