Graph distances of continuum long-range percolation

Abstract

We consider a version of continuum long-range percolation on finite boxes of ${\mathbb{R}}^d$ 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 $r^{-s}$ 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 $s=d$ and $s=2d$ in analogy to classical long-range percolation on ${\mathbb{Z}}^d$, by using techniques which are based on an analysis of the underlying Poisson point process.

References in corpus (5)

• On the scaling of the chemical distance in long-range percolation models
• A lower bound for the chemical distance in sparse long-range percolation models
• Distances in critical long range percolation
• Sharp asymptotic for the chemical distance in long-range percolation
• The random walk on the random connection model

Cited by in corpus (1)

• The longest edge in discrete and continuous long-range percolation