Expansion of the Critical Intensity for the Random Connection Model

TL;DR

The paper analyzes continuum percolation in the Random Connection Model by developing a lace expansion for the two-point function and establishing an asymptotic expansion of the critical intensity $\lambda_c$ as the ambient dimension $d$ grows large. The expansion expresses $\lambda_c$ in terms of convolutions $\varphi^{\star n}(0)$ and a hierarchy of lace-expansion coefficients, with rigorous upper and lower bounds for the leading terms. The authors apply the general result to four kernels—the Gilbert disk / Hyper-Sphere, Hyper-Cube, Gaussian, and Coordinate-wise Cauchy—deriving explicit high-dimensional expansions (often exponentially decaying in $d$) for each case. A key methodological contribution is the precise control of lace-expansion coefficients via Mecke’s equation and the BK inequality, together with diagrammatic (graphical) notation for complex convolution structures. The results highlight a marked contrast with lattice percolation, where corrections decay algebraically in $d$, and provide a robust framework for understanding high-dimensional continuum percolation phenomena with potential extensions to marked models.

Abstract

We derive an asymptotic expansion for the critical percolation density of the random connection model as the dimension of the encapsulating space tends to infinity. We calculate rigorously the first expansion terms for the Gilbert disk model, the hyper-cubic model, the Gaussian connection kernel, and a coordinate-wise Cauchy kernel.

Figures (7)

• Figure 1: Left: The Hyper-Sphere RCM -- two Poisson points are connected whenever the circles of radius $R/2$ overlap. Right: The Hyper-Cube RCM -- two Poisson points are connected whenever the cubes of side length $L/2$ overlap.
• Figure 2: Diagrams of the $\psi_0$, $\psi$, and $\psi_n$ functions.
• Figure 3: Diagrams of the $\overline{\psi}_0$, $\overline{\psi}$, and $\overline{\psi}_n$ functions, which we use to bound the $\psi_0$, $\psi$, and $\psi_n$ functions.
• Figure 4: Sketch of $\widehat{\varphi}\left(k\right)$ against $\lvert*\rvert{k}$. It approaches its maximum quadratically as $\lvert*\rvert{k}\to0$. The first local maximum of $J_{\frac{d}{2}}$ occurs at $j'_{\frac{d}{2},1}\sim \frac{d}{2}+\gamma_1\left(\frac{d}{2}\right)^\frac{1}{3}$. The first zero of $\widehat{\varphi}\left(k\right)$ occurs at $\lvert*\rvert{k}R(d) = j_{\frac{d}{2},1}\sim \frac{d}{2}+\gamma_2\left(\frac{d}{2}\right)^\frac{1}{3}$ where $\gamma_2>\gamma_1$. Furthermore, $\widehat{\varphi}\left(k\right)$ is strictly decreasing until $\lvert*\rvert{k}R(d) = j_{\frac{d}{2}+1,1}\sim \frac{d}{2}+\gamma_2\left(\frac{d}{2}\right)^\frac{1}{3} + 1$.
• Figure 5: Plots of $\frac{1}{d}\log\left(\cdot\right)$ for each of the diagrams for the Hyper-Sphere RCM. For comparison, $\frac{1}{d}\log\varphi^{\star1\star2\cdot3}\left(\mathbf{0}\right)$ is represented in both plots - it is the smallest of the larger diagrams and the largest of the smaller diagrams in the higher dimensions.

Theorems & Definitions (83)

• Remark 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Corollary 1.7
• Corollary 1.8
• Corollary 1.9
• Remark 1.10