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2009 Interlacement percolation on transient weighted graphs
Augusto Teixeira
Author Affiliations +
Electron. J. Probab. 14: 1604-1627 (2009). DOI: 10.1214/EJP.v14-670

Abstract

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value $u_*$ for the percolation of the vacant set is finite. We also prove that, once $\mathcal{G}$ satisfies the isoperimetric inequality $I S_6$ (see (1.5)), $u_*$ is positive for the product $\mathcal{G} \times \mathbb{Z}$ (where we endow $\mathbb{Z}$ with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value $u_*$.

Citation

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Augusto Teixeira. "Interlacement percolation on transient weighted graphs." Electron. J. Probab. 14 1604 - 1627, 2009. https://doi.org/10.1214/EJP.v14-670

Information

Accepted: 9 July 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1192.60108
MathSciNet: MR2525105
Digital Object Identifier: 10.1214/EJP.v14-670

Subjects:
Primary: 60K35
Secondary: 82C41

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