Supercritical phase of the random connection model

Given $d \in {\bf N}, \lambda >0$, the random connection model in a region $A \subseteq {\bf R}^d$ is a graph with vertex set given by a homogeneous Poisson point process of intensity $\lambda$ in $A$, with an edge placed between each pair $x,y$ of vertices with probability $\phi(\|x-y\|)$, where $\phi: {\bf R}_+ \to [0,1]$ is a nonincreasing finite-range connection function. We show that if $d \geq 3$ and $\lambda$ is strictly supercritical for $A = {\bf R}^d$, then the model remains supercritical if it is restricted to a region $A$ of the form ${\bf R}^2 \times [-K/2,K/2]^{d-2}$, 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.