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Sharpness of the phase transition for continuum percolation in R2
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Matematica Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
Univ Geneva, 2-4 Rue Lievre, CH-1211 Geneva, Switzerland.
Inst Matematica Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
2018 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 172, no 1-2, p. 525-581Article in journal (Refereed) Published
Abstract [en]

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter.c. Moreover, For. <.c, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution. At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius. For. >.c, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast. The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

Place, publisher, year, edition, pages
2018. Vol. 172, no 1-2, p. 525-581
Keywords [en]
Percolation, Poisson point processes, Critical behavior, Sharp thresholds
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-366947DOI: 10.1007/s00440-017-0815-8ISI: 000443981900011OAI: oai:DiVA.org:uu-366947DiVA, id: diva2:1266571
Funder
Swedish Research Council, 637-2013-7302Available from: 2018-11-28 Created: 2018-11-28 Last updated: 2018-11-28Bibliographically approved

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Ahlberg, Daniel

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