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The critical probability for random Voronoi percolation in the plane is 1/2. (English) Zbl 1100.60054
Summary: We study percolation in the following random environment: let \(Z\) be a Poisson process of constant intensity on \(\mathbb R^{2}\), and form the Voronoi tessellation of \(\mathbb R^{2}\) 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 \(\mathbb Z^{2}\).
The result corresponding to Harris’ theorem for bond percolation in \(\mathbb Z^{2}\) 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 \(\mathbb Z^{2}\); we hope they will be applicable in other contexts as well.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
52C99 Discrete geometry
60C05 Combinatorial probability
82B43 Percolation
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