Probability Theory and Related Fields

, Volume 136, Issue 3, pp 417–468

The critical probability for random Voronoi percolation in the plane is 1/2

  • Béla Bollobás
  • Oliver Riordan
Article

DOI: 10.1007/s00440-005-0490-z

Cite this article as:
Bollobás, B. & Riordan, O. Probab. Theory Relat. Fields (2006) 136: 417. doi:10.1007/s00440-005-0490-z

Abstract

We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ2, and form the Voronoi tessellation of ℝ2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p>1/2 then the union of the black cells contains an infinite component with probability 1, while if p<1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in ℤ2.

The result corresponding to Harris' Theorem for bond percolation in ℤ2 is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for ℤ2; we hope they will be applicable in other contexts as well.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Béla Bollobás
    • 1
    • 2
  • Oliver Riordan
    • 2
    • 3
  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  2. 2.Trinity CollegeCambridgeUK
  3. 3.Royal Society Research Fellow, Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeUK