Invariant percolation and harmonic Dirichlet functions

By Damien Gaboriau. 2004

67 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3ef2b8c02626594b005903abc133a222289fe044

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Citations in the corpus

Orbit Equivalence and Measured Group Theory (2010, background): “For instance, in connection with group l2-Betti numbers β((2)) n (Γ), this was useful to attack: – various vanishing results in [Gab02, ST07]; – the study of harmonic Dirichlet functions on percolation subgraphs [Gab05b]; – the comparison between the uniform isoperimetric constant and β((2)) 1 (Γ) [”
Random Complexes and l^2-Betti Numbers (2008, background): “24), which depends on a definition of Gaboriau (2005). Now we extend the identity to higher dimensions.”
Percolation Theory (2005, unknown): “In the general case, it was recently proved by Gaboriau that if the graph G is unimodular, transitive, locally finite and supports non-constant harmonic Dirichlet functions (i.e. harmonic functions whose gradient is in ℓ2), then indeed pc(G ) pu(G ).”
On limits of finite graphs (2005, background): “eafgraph is isomorphic to A. We say that G is the limit graphing of the weakly convergent sequence {G n}∞ n=1 , if for any r ≥ 1 and A ∈ U r,d lim n→∞ p Gn (A) = pG(A). Combining the ideas of [2] and [3] one can see that for any weakly convergent graph sequence there exists a measurable limit graphi”
A measurable-group-theoretic solution to von Neumann’s problem (2007, methodology): “We define the following equivalence subrelation on X (see [ Gab05 , Sect.”