Invariant percolation and harmonic dirichlet functions
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In this paper, the existence of the non-uniqueness phase for the Bernoulli percolation on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions was shown.Abstract:
The main goal of this paper is to answer question~1.10 and settle conjecture~1.11 of Benjamini-Lyons-Schramm \cite{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.read more
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On the random-cluster model: I. Introduction and relation to other models
C.M. Fortuin,P.W. Kasteleyn +1 more
TL;DR: It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.
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J. Feldman,Calvin C. Moore +1 more
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Density and uniqueness in percolation
Robert M. Burton,M. Keane +1 more
TL;DR: In this paper, two results on site percolation on the d-dimensional lattice, d≧1 arbitrary, are presented, and they extend to a broad class of finite-dimensional models.
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