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 > p_c(G) then there exists a positive constant c_p such thatPp(n≤|K|<∞)≤e-cpn\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf {P}_p(n \\le |K| < \\infty ) \\le e^{-c_p n} \\end{aligned}$$\\end{document}for every nge 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.
Highlights
Let G = (V, E) be a connected, locally finite graph
We develop a theory of supercritical percolation on nonamenable transitive graphs, studying in particular the distribution of finite clusters, the geometry of the infinite clusters, and the regularity of the dependence of various observables on p
For supercritical percolation on unimodular transitive nonamenable graphs, positive speed of the random walk was already proven to hold by Benjamini et al [10, Theorem 1.3]
Summary
Note that the theorem fails without the assumption of transitivity: For example, the graph obtained by attaching a 3-regular tree and a 4-regular tree by a single edge between their respective origins has pc = 1/3 and ζ (1/2) = 0 It was observed in [8] that the argument of [2] can be generalized to prove that ζ ( p) = 0 for every p > pc whenever G is a Cayley graph of a finitely presented amenable group. In Corollary 3.5 we prove via a different argument that ζ ( p) = 0 for every amenable transitive graph and every pc ≤ p < 1 This gives a converse to Theorem 1.1, so that, combining both results, we obtain the following appealing percolation-theoretic characterization of nonamenability for transitive graphs: Corollary 1.2 Let G be a connected, locally finite, transitive graph. We note in particular that Theorem 1.1 and Corollary 1.2 resolve several questions raised by Bandyopadhyay et al [8, Questions 1 and 2 and Conjecture 1]
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