# Bond percolation with attenuation in high dimensional Voronǒ tilings

## Abstract

Let P be the set of points in a realization of a uniform Poisson process in Rn. The set P determines a Voronǒ tiling of Rn. Construct an infinite graph G with vertex set P and edges joining vertices when the corresponding Voronǒ cells share a (n-1)-dimensional boundary face. We consider bond percolation models on G obtained by declaring each edge xy of 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(5n logn)-1 ≤ pc(n) ≤ C2-n√nlogn. 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) = (loge 2 + o(1))n/2rn, where rn is the radius of the n-dimensional sphere that, on average, contains a single point of P. © 2009 Wiley Periodicals, Inc.

## Publication Title

Random Structures and Algorithms

## Recommended Citation

Balister, P., & Bollobás, B.
(2010). Bond percolation with attenuation in high dimensional Voronǒ tilings.* Random Structures and Algorithms**, 36* (1), 5-10.
https://doi.org/10.1002/rsa.20295