Spread-out percolation in R^d

Details Spread-out percolation in R^d Spread-out percolation in R^d

2005-08-23

arxiv.org/abs/math/0508430v2

Random Struct. Algorithms 31 (2007), 239-246.

Mathematics

Probability

9 pages. Title changed. Minor changes to text, including updated references to [3]. To appear in Random Structures and Algorit

Scientific paper

10.1002/rsa.20175

Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$ within distance $r$ independently with probability $p$. This is the simplest case of a `spread-out' percolation model studied by Penrose, who showed that, as $r\to\infty$, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of the authors on inhomogeneous random graphs.

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