  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.
