Electronic Communications in Probability

A short proof of the phase transition for the vacant set of random interlacements

Balázs Ráth

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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 $d \geq 3$.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 3, 11 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320930

Digital Object Identifier
doi:10.1214/ECP.v20-3734

Mathematical Reviews number (MathSciNet)
MR3304409

Zentralblatt MATH identifier
1307.60146

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Percolation Random Interlacements

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Ráth, Balázs. A short proof of the phase transition for the vacant set of random interlacements. Electron. Commun. Probab. 20 (2015), paper no. 3, 11 pp. doi:10.1214/ECP.v20-3734. https://projecteuclid.org/euclid.ecp/1465320930


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References

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