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Pré-Publication, Document De Travail Année : 2005

Invariant Percolation and Harmonic Dirichlet Functions

Résumé

The main goal of this paper is to answer question~1.10 and settle conjecture~1.11 of Benjamini-Lyons-Schramm \cite{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.
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Dates et versions

hal-00001606 , version 1 (24-05-2004)
hal-00001606 , version 2 (09-03-2005)
hal-00001606 , version 3 (25-05-2005)

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Damien Gaboriau. Invariant Percolation and Harmonic Dirichlet Functions. 2005. ⟨hal-00001606v2⟩
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