Percolation In The Hyperbolic Plane (Extended Abstract)

ITAI BENJAMINI AND ODED SCHRAMM The Voronoi model for percolation in H 2 . Percolation has been studied extensively in R d and in lattices of R d . In [1], we have proposed some conjectures about percolation in quite general settings. Here, we begin the investigation of percolation in hyperbolic space, and concentrate on dimension 2. It turns out that percolation in hyperbolic space is very different from percolation in Euclidean geometry. The difference is reminiscent, and indeed analogous, to the similar disparity of the behaviour of Brownian motion in the two settings. Let us start by describing the Voronoi percolation model, which seems to be more suitable than lattice models or the lily-pad model. We have recently found that the Voronoi model is useful in the study of the conformal invariance conjecture for critical percolation. Fix two parameters, ? 0; p 2 [0; 1]. Let Xw be a Poisson point process in the hyperbolic plane H 2 , with density p, and let X b be an independent..