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2015 A short proof of the phase transition for the vacant set of random interlacements
Balázs Ráth
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Electron. Commun. Probab. 20: 1-11 (2015). DOI: 10.1214/ECP.v20-3734

Abstract

The vacant set of random interlacements at level $u>0$, introduced in [Sznitman 2009], is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories, where $u$ is a parameter controlling the density of the cloud. It was proved in [Sznitman 2009] and [Sidoravicius, Sznitman 2010] that for any $d \geq 3$ there exists a positive and finite threshold $u_*$ such that if $u<u_*$ then the vacant set percolates and if $u>u_*$ then the vacant set does not percolate. We give an elementary proof of these facts. Our method also gives simple upper and lower bounds on the value of $u_*$ for any $d \geq 3$.

Citation

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Balázs Ráth. "A short proof of the phase transition for the vacant set of random interlacements." Electron. Commun. Probab. 20 1 - 11, 2015. https://doi.org/10.1214/ECP.v20-3734

Information

Accepted: 7 January 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1307.60146
MathSciNet: MR3304409
Digital Object Identifier: 10.1214/ECP.v20-3734

Subjects:
Primary: 60K35
Secondary: 82B43

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