Exponential decay of connection probabilities for subcritical Voronoi percolation in \(\mathbb {R}^d\)

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Abstract

We prove that for Voronoi percolation on \(\mathbb {R}^d\) \((d\ge 2)\), there exists \(p_c=p_c(d)\in (0,1)\) such that

  • for \(p<p_c\), there exists \(c_p>0\) such that \(\mathbb {P}_p[0\text { connected to distance }n]\le \exp (-c_pn)\),

  • there exists \(c>0\) such that for \(p>p_c, \mathbb {P}_p[0\text { connected to infinity}]\ge c(p-p_c)\).

For dimension 2, this result offers a new way of showing that \(p_c(2)=1/2\). This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition; see [10, 11].

Mathematics Subject Classification

60K35 82B21 82B43 
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Notes

Acknowledgements

The authors are thankful to Asaf Nachmias for reading the manuscript and his helpful comments. This research was supported by the IDEX grant from Paris-Saclay, a grant from the Swiss FNS, and the NCCR SwissMAP.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Hugo Duminil-Copin
    • 1
    • 2
  • Aran Raoufi
    • 2
  • Vincent Tassion
    • 3
  1. 1.Université de GenèveGenevaSwitzerland
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  3. 3.ETH ZurichZurichSwitzerland

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