Invariant Percolation and Harmonic Dirichlet Functions

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The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those of the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of $\ell^2$ Betti numbers.
to appear in Geometric And Functional Analysis (GAFA)

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Probability, Dynamical Systems, Group Theory, MSC: 60K35, 82B43, 31C05, 37A20, 05C25, 05C80, 37R30

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http://salesiana.dossiersoluciones.com/handle/123456789/71301

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0405458
 http://arxiv.org/abs/math/0405458

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arXiv