Bond percolation with attenuation in high dimensional Voronoĭ tilings

Let $\cal{P}$ be the set of points in a realization of a uniform Poisson process in ℝ n . The set $\cal{P}$ determines a Voronoĭ tiling of ℝ n . Construct an infinite graph $\cal{G}$ with vertex set $\cal{P}$ and edges joining vertices when the corresponding Voronoĭ cells share a ( n − 1)‐dimensional boundary face. We consider bond percolation models on $\cal{G}$ obtained by declaring each edge x y of $\cal{G}$ open independently with probability p (∥ x − y ∥), depending only on the Euclidean distance ∥ x − y ∥ between the vertices. We give some sufficient conditions on p ( t ) that ensures that an infinite connected component (i.e., percolation) occurs, or does not occur. In particular, we show that for p ( t ) = p is a constant, there is a phase transition at a critical probability p = p c ( n ), where 2 − n (5 n log n ) −1 ≤ p c ( n ) ≤ C 2 − n $\sqrt{n}\;{\rm log} \; n$ . We also show that if p ( t ) = e − λ t then there is a phase transition at a critical parameter λ = λ c ( n ), where λ c ( n ) = (log e 2 + o (1)) n /2 r n , where r n is the radius of the n ‐dimensional sphere that, on average, contains a single point of $\cal{P}$ . © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010