Graph distances of continuum long-range percolation

Abstract

We consider a version of continuum long-range percolation on finite boxes of ${\mathbb{R}}^{\mathit{d}}$ in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance r is connected with probability proportional to ${\mathit{r}}^{-\mathit{s}}$ for a certain constant s. We explore the graph-theoretical distance in this model. The aim of this paper is to show that this random graph model undergoes phase transitions at values $\mathit{s}=\mathit{d}$ and $\mathit{s}=2\mathit{d}$ in analogy to classical long-range percolation on ${\mathbb{Z}}^{\mathit{d}}$, by using techniques which are based on an analysis of the underlying Poisson point process.