Supercritical phase of the random connection model

Abstract

Given $d\in \mathbf{N},\hat{I}»>0$, the random connection model in a region $A\subseteq {\mathbf{R}}^d$ is a graph with vertex set given by a homogeneous Poisson point process of intensity $\hat{I}»$ in $A$, with an edge placed between each pair $x,y$ of vertices with probability $\ddot{I}•(\Vert x-y\Vert )$, where $\ddot{I}•:{\mathbf{R}}_{+}\to [0,1]$ is a nonincreasing finite-range connection function. We show that if $d\ge 3$ and $\hat{I}»$ is strictly supercritical for $A={\mathbf{R}}^d$, then the model remains supercritical if it is restricted to a region $A$ of the form ${\mathbf{R}}^2\times [-K/2,K/2{]}^{d-2}$, provided $K$ is sufficiently large. This is a continuum analogue of a well-known result of Grimmett and Marstrand for lattice percolation. We prove this by adapting Grimmett and Marstrand's original proof; Faggionato and Hartarsky have also proved this recently by other means.

21 pages, 2 figures