×

zbMATH — the first resource for mathematics

Existence of an unbounded vacant set for subcritical continuum percolation. (English) Zbl 1401.60173
Summary: 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 \(d\geq 2\), finite moment of order \(d\) is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] D. Ahlberg, V. Tassion, and A. Teixeira. Sharpness of the phase transition in continuum percolation in \(\mathbb{R} ^2\). Probab. Theory Related Fields, to appear. · Zbl 1404.60143
[2] B. Bollobás and O. Riordan. The critical probability for random Voronoi percolation in the plane is 1/2. Probab. Theory Related Fields, 136(3):417–468, 2006. · Zbl 1100.60054
[3] J.-B. Gouéré. Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab., 36(4):1209–1220, 2008. · Zbl 1148.60077
[4] P. Hall. On continuum percolation. Ann. Probab., 13(4):1250–1266, 1985. · Zbl 0588.60096
[5] H. Kesten. The critical probability of bond percolation on the square lattice equals \(1\over 2\). Comm. Math. Phys., 74(1):41–59, 1980. · Zbl 0441.60010
[6] T. M. Liggett, R. H. Schonmann, and A. M. Stacey. Domination by product measures. Ann. Probab., 25(1):71–95, 1997. · Zbl 0882.60046
[7] M. D. Penrose. Non-triviality of the vacancy phase transition for the Boolean model. Preprint, see arXiv:1706.02197. · Zbl 1394.60101
[8] R. Roy. The Russo-Seymour-Welsh theorem and the equality of critical densities and the “dual” critical densities for continuum percolation on \(\textbf{R}^2\). Ann. Probab., 18(4):1563–1575, 1990. · Zbl 0719.60119
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.