Conformal Invariance of Voronoi Percolation

Abstract:

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 μ, and a Riemannian metric ds. Points are scattered according to a Poisson point process on (M,μ), with some density λ. 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 λ→∞, if the metric ds is replaced by any (smoothly) conformal metric ds'. Additionally, it is conjectured that if μ and μ' are two measures comparable to the Riemannian volume measure, then replacing μ by μ' does not effect the limiting crossing probabilities.