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2018 Existence of an unbounded vacant set for subcritical continuum percolation
Daniel Ahlberg, Vincent Tassion, Augusto Teixeira
Electron. Commun. Probab. 23: 1-8 (2018). DOI: 10.1214/18-ECP152


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.


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Daniel Ahlberg. Vincent Tassion. Augusto Teixeira. "Existence of an unbounded vacant set for subcritical continuum percolation." Electron. Commun. Probab. 23 1 - 8, 2018.


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

zbMATH: 1401.60173
MathSciNet: MR3863919
Digital Object Identifier: 10.1214/18-ECP152

Primary: 60G55 , 60K35 , 82B43

Keywords: dependent environments , percolation , phase transition


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