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

By Béla Bollobás, Oliver Riordan. 2004

178 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/59c5c655986b2d772f33af041d8b3702d2858be5

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Citations in the corpus

The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 (2010, methodology): “The previous two sections combine in order to provide estimates on crossing probabilities (see [5, 6] for applications in the case of percolation).”
Crossing probabilities for Voronoi percolation (2014, methodology): “The ﬁrst proof of this, given by [BR06], required some RSW-like bounds.”
A Short Proof of the Harris–Kesten Theorem (2004, result): “Other proofs of their results have since been developed, for example that due to Zhang (see [10]), and very much more general results are now known; however, none of the proofs is very simple.”
Concentration for self-bounding functions and an inequality of Talagrand (2006, methodology): “The same technicality arises in the Voronoi setting; see Section 8 of [ 4 ].”
Percolation of random nodal lines (2016, unknown): “and a fundamental ingredient in the obtention of Schramm-Loewner Evolution as a scaling limit [39,14]. Russo-Seymour-Welsh theory has been extended to dependent models [15], as well as to some random [12,40] or non-planar [32,6] lattices. It remains an open problem to generalize it to triangulations”
Conformally invariant scaling limits: an overview and a collection of problems (2006, methodology): “Notable recent progress has been made for Voronoi percolation by Bollobás and Riordan [15], who established the very useful Russo-Seymour-Welsh theorem, as well as pc = 1/2 for Voronoi percolation.”
Influence and sharp-threshold theorems for monotonic measures (2005, methodology): “We next use an argument taken from [4, 5] to obtain a more locally defined event.”