The Triangle Condition for the Marked Random Connection Model

TL;DR

The authors analyze a marked continuum percolation model (RCM) in which vertices form a marked Poisson process and edges appear with a probability depending on spatial distance and marks. They develop a lace-expansion framework for Poisson processes and translate it into an operator form, proving an infrared bound and the triangle condition in high dimension ($d\ge d^*$, with $d^*>6$). This yields mean-field critical exponents for a large class of dependent percolation models, extending the triangle-condition paradigm beyond independence. They verify assumptions via concrete model families (finite-variance, space-mark factorisation, multivariate Gaussian marks, and bounded-radius Boolean discs) and establish uniform convergence to an Ornstein–Zernike equation at criticality. The results provide a rigorous link between dependency structures in marked continuum graphs and universal mean-field critical behavior, with implications for percolation thresholds and critical exponents in complex networks.

Abstract

We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $\mathbb{R}^d$. Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks. Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely. Our focus is on the transition between the low- and high-density phase, where the system is critical. We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid. This implies the triangle condition, and thus mean-field behaviour. We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates.

Figures (6)

• Figure 1: Diagrams of the $\psi_0$, $\psi$, and $\psi_n$ functions.
• Figure 2: Diagrams of the $\overline{\psi}_0$, and $\overline{\psi}$ functions.
• Figure 3: Diagrams representing $\overline{W_k}$ and $\overline{H_k}$. In this figure, the marks on edges incident to filled square vertices need not be equal.
• Figure 4: If $\widehat{\mathcal{T}}_\lambda(k)$ and $\widehat{a} (k)$ commute, then they are simultaneously diagonalizable in the sense of Theorem \ref{['thm:spectraltheorem']}. The arguments of their diagonal functions ($\widetilde{\tau}_\lambda(k)$ and $\widetilde{a}(k)$ respectively) are related by the monotone increasing function $x\mapsto \tfrac{x}{1-\lambda x}$ depicted here.
• Figure 5: Sketch of $\widehat{\varphi}\left(k;a,b\right)$ 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 $\widehat{\varphi}\left(k;a,b\right)$ occurs at $\lvert*\rvert{k}R_{a,b} = j_{\frac{d}{2},1}\sim \frac{d}{2}+\gamma_2\left(\frac{d}{2}\right)^\frac{1}{3}$ where $\gamma_2>\gamma_1$. Furthermore, $\widehat{\varphi}\left(k;a,b\right)$ is strictly decreasing until $\lvert*\rvert{k}R_{a,b} = j_{\frac{d}{2}+1,1}\sim \frac{d}{2}+\gamma_2\left(\frac{d}{2}\right)^\frac{1}{3} + \frac{1}{2}$.

Theorems & Definitions (116)

• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 2.5
• Corollary 2.6
• Remark 2.7
• Theorem 3.1: BK inequality
• Lemma 3.2: Bounds on the Operator Norm