Conformal Invariance of Voronoi Percolation

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 deene Voronoi percolation on a manifold M , you need a measure , and a Riemannian metric ds. Points are scattered according to a Poisson point process on (M;), with some density. Each cell in the Voronoi tessellation determined by the chosen points is declared open with some xed 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 ! 1, if the metric ds is replaced by any (smoothly) conformal metric ds 0. Additionally, it is conjectured that if and 0 are two measures comparable to the Riemannian volume measure, then replacing by 0 does not eeect the limiting crossing probabilities.