Electronic Communications in Probability

Existence of an unbounded vacant set for subcritical continuum percolation

Daniel Ahlberg, Vincent Tassion, and Augusto Teixeira

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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

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

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

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 60G55: Point processes

percolation phase transition dependent environments

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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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