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 c onformal change of metric. To define Voronoi percolation on a manifold M, you need a measure mu, and a Riemannian metric ds. Points are scat tered according to a Poisson point process on (M, mu), with some densi ty lambda. Each cell in the Voronoi tessellation determined by the cho sen points is declared open with some fixed probability p, and closed with probability 1 - p, independently of the other cells. The above co nformal invariance statement means that under certain conditions, the probability for an open crossing between two sets is asymptotically un changed, as lambda --> infinity, if the metric ds is replaced by any ( smoothly) conformal metric ds'. Additionally, it is conjectured that i f mu and mu' are two measures comparable to the Riemannian volume meas ure, then replacing mu by mu' does not effect the limiting crossing pr obabilities.