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Journal Articles Geometric And Functional Analysis Year : 2005

## Invariant Percolation and Harmonic Dirichlet Functions

Damien Gaboriau

#### Abstract

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.

### Dates and versions

hal-00001606 , version 1 (24-05-2004)
hal-00001606 , version 2 (09-03-2005)
hal-00001606 , version 3 (25-05-2005)

### Identifiers

• HAL Id : hal-00001606 , version 3
• ARXIV :
• DOI :

### Cite

Damien Gaboriau. Invariant Percolation and Harmonic Dirichlet Functions. Geometric And Functional Analysis, 2005, 15 (5), pp.1004-1051. ⟨10.1007/s00039-005-0539-2⟩. ⟨hal-00001606v3⟩

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