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



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 



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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