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Sharpness of the phase transition for continuum percolation in \(\mathbb {R}^2\)

Abstract

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 \(\lambda _c\). Moreover,

  • For \(\lambda < \lambda _c\), 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 \(\lambda > \lambda _c\), 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.

Graphical Abstract

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Acknowledgements

We would like to thank Caio Teodoro for the careful reading, suggestions and corrections. This work began during a visit of V. T. to IMPA, that he thanks for support and hospitality. We thank the Centre Intradisciplinaire Bernoulli (CIB) and Stardû for hosting the authors. D. A. was during the course of this project financed by Grant 637-2013-7302 from the Swedish Research Council. A. T. is grateful to CNPq for its financial contribution to this work through the Grants 306348/2012-8, 478577/2012-5 and 309356/2015-6 and FAPERJ through Grant Number 202.231/2015. V. T. acknowledges support from the Swiss NSF.

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Ahlberg, D., Tassion, V. & Teixeira, A. Sharpness of the phase transition for continuum percolation in \(\mathbb {R}^2\) . Probab. Theory Relat. Fields 172, 525–581 (2018). https://doi.org/10.1007/s00440-017-0815-8

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  • DOI: https://doi.org/10.1007/s00440-017-0815-8

Keywords

  • Percolation
  • Poisson point processes
  • Critical behavior
  • Sharp thresholds

Mathematics Subject Classification

  • 60K35
  • 82B43
  • 60G55