Invariant percolation and harmonic Dirichlet functions

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 l(2) Betti numbers.