## Electronic Communications in Probability

### Existence of an unbounded vacant set for subcritical continuum percolation

#### Abstract

We consider the Poisson Boolean percolation model in $\mathbb{R} ^2$, where the radius of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R} ^d$, for any $d\ge 2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 63, 8 pp.

Dates
Received: 26 June 2017
Accepted: 16 July 2018
First available in Project Euclid: 15 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1536977436

Digital Object Identifier
doi:10.1214/18-ECP152

Mathematical Reviews number (MathSciNet)
MR3863919

Zentralblatt MATH identifier
1401.60173

#### Citation

Ahlberg, Daniel; Tassion, Vincent; Teixeira, Augusto. Existence of an unbounded vacant set for subcritical continuum percolation. Electron. Commun. Probab. 23 (2018), paper no. 63, 8 pp. doi:10.1214/18-ECP152. https://projecteuclid.org/euclid.ecp/1536977436

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