Supercritical phase of the random connection model

TL;DR

This paper demonstrates that the supercritical phase of the random connection model persists in certain lower-dimensional regions when the region's width is sufficiently large, extending a known lattice percolation result to continuum models.

Contribution

It extends the supercritical phase stability result from infinite space to specific finite-width regions in the continuum setting, adapting the lattice percolation proof.

Findings

Supercritical phase persists in regions of the form R^2 x [-K/2,K/2]^{d-2} for large K.

The proof adapts Grimmett and Marstrand's lattice percolation argument to the continuum model.

The result applies for dimensions d ≥ 3 and supercritical intensity λ.

Abstract

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.