2004/10/31 by Béla Bollobás, Bela Bollobas, Oliver Riordan · 3 citations
Mathematics · #Centroidal Voronoi tessellation #Combinatorics #Continuum percolation theory #Critical exponent #Discrete mathematics #Geometry #Markov Chains and Monte Carlo Methods #Mathematical proof #Mathematics #Percolation (cognitive psychology) #Percolation critical exponents #Physics #Plane (geometry) #Point process #Point processes and geometric inequalities #Poisson distribution #Poisson point process #Probability theory #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Voronoi diagram #math.PR #msc:60K35 #msc:82B43
paper · pdf · doi:10.1007/s00440-005-0490-z
published as Probability Theory and Related Fields 136 (2006), pp 417--468. · 55 pages, 20 figures; minor changes; to appear in Probability Theory and Related Fields
arxiv created 2005/09/27 · openalex publication_date 2005/12/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study percolation in the following random environment: let Z be a Poisson process of constant intensity in the plane, and form the Voronoi tessellation of the plane 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 the square lattice. The result corresponding to Harris' Theorem for bond percolation in the square lattice 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 the square lattice; we hope they will be applicable in other contexts as well.