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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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