Abstract.
The main goal of this paper is to answer Question 1.10 and settle Conjecture 1.11 of Benjamini–Lyons–Schramm [BenLS] relating harmonic Dirichlet functions on a graph to those on 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 ℓ2 Betti numbers.
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Received: May 2004 Revised: March 2005 Accepted: May 2005
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Gaboriau, D. Invariant percolation and harmonic Dirichlet functions. GAFA, Geom. funct. anal. 15, 1004–1051 (2005). https://doi.org/10.1007/s00039-005-0539-2
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DOI: https://doi.org/10.1007/s00039-005-0539-2