Invariant Percolation and Harmonic Dirichlet Functions
نویسنده
چکیده
The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of BenjaminiLyons-Schramm [BLS99] 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 Betti numbers. Mathematical Subject Classification: 60K35, 82B43, 31C05, 37A20, 05C25, 05C80, 37R30
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تاریخ انتشار 2004