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Invariant Percolation and Harmonic Dirichlet Functions

Damien Gaboriau
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Abstract

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.

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Damien Gaboriau. 2005-05-25. Invariant Percolation and Harmonic Dirichlet Functions. https://arxiv.org/abs/math/0405458

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