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Conformal invariance of Voronoi percolation. (English) Zbl 0921.60081
Summary: It is proved that in the Voronoi model for percolation in dimension 2 and 3, the crossing probabilities are asymptotically invariant under conformal change of metric. To define Voronoi percolation on a manifold \(M\), you need a measure \(\mu\), and a Riemannian metric \(ds\). Points are scattered according to a Poisson point process on \((M,\mu)\), with some density \(\lambda\). Each cell in the Voronoi tessellation determined by the chosen points is declared open with some fixed probability \(p\), and closed with probability \(1-p\), independently of the other cells. The above conformal invariance statement means that under certain conditions, the probability for an open crossing between two sets is asymptotically unchanged, as \(\lambda\to\infty\), if the metric \(ds\) is replaced by any (smoothly) conformal metric \(ds'\). Additionally, it is conjectured that if \(\mu\) and \(\mu'\) are two measures comparable to the Riemannian volume measure, then replacing \(\mu\) by \(\mu'\) does not effect the limiting crossing probabilities.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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