Bond percolation with attenuation in high dimensional Voronoĭ tilings

Abstract

AbstractLet $\cal{P}$ be the set of points in a realization of a uniform Poisson process in ℝn. The set $\cal{P}$ determines a Voronoĭ tiling of ℝn. 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 xy 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 = pc(n), where 2−n(5nlog n)−1 ≤ pc(n) ≤ C2−n $\sqrt{n}\;{\rm log} \; n$. We also show that if p(t) = e−λt then there is a phase transition at a critical parameter λ = λc(n), where λc(n) = (log e2 + o(1))n/2rn, where rn 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