Mean-field behaviour of the random connection model on hyperbolic space

TL;DR

This work analyzes the random connection model on hyperbolic spaces ${\mathbb{H}^d}$ for $d=2,3$, establishing a nontrivial percolation phase transition and proving mean-field critical exponents. The authors develop a half-space methodology linked to hyperbolic isoperimetry to bound susceptibility, magnetisation, and cluster tails, showing $\gamma=1$, $\beta=1$, $\delta=2$, and $\Delta=2$, and that $\lambda_c=\lambda_T$. They also prove a near-critical tail${}^{-1/2}$ behavior for the cluster size distribution and derive cluster-moment bounds via Greg-tree machinery, all under a long-distance assumption and a finite positive integral of the connection kernel. Unlike Euclidean high-dimensional results that rely on triangle diagrams, the proofs here exploit hyperbolic geometry and stepping-stone constructions, with detailed half-space proofs in ${\mathbb{H}^2}$ and ${\mathbb{H}^3}$. The findings connect continuum hyperbolic percolation to mean-field behavior observed in hyperbolic random graphs, while highlighting the geometric mechanisms that underpin these universal exponents.

Abstract

We study the random connection model on hyperbolic space $\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $λ>0$. Upon variation of $λ$ there is a percolation phase transition: there exists a critical value $λ_c>0$ such that for $λ<λ_c$ all clusters are finite, but infinite clusters exist for $λ>λ_c$. We identify certain critical exponents that characterize the clusters at (and near) $λ_c$, and show that they agree with the mean-field values for percolation. We derive the exponents through isoperimetric properties of critical percolation clusters rather than via a calculation of the triangle diagram.

Figures (7)

• Figure 1: Simulations of the random connection model on the Poincaré disc model of ${\mathbb{H}^2}$ with $\varphi\left(x,y\right) = \exp\left(-2\mathrm{dist}\left(x,y\right)\right)$ and different intensities.
• Figure 2: Sketch showing construction of the length \ref{['eqn:avoidancelength']} in the Poincaré disc model of ${\mathbb{H}^2}$.
• Figure 3: Sketch of the set $\mathcal{M}_\varepsilon(H,x)$ in the Poincaré Disc model for ${\mathbb{H}^2}$.
• Figure 4: Sketch of the 'stepping stones' used to connect $o$ to a far half-space in Section \ref{['sec:SusceptibilityUpperBound']}.
• Figure 5: Sketch of the 'stepping stones' used to connect $o$ to two disjoint far half-spaces in Section \ref{['sec:PercolationUpperBounds']}.

Theorems & Definitions (66)

• Proposition 1.0
• Theorem 1.1: Critical exponents
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• proof : Proof of the lower bound in Theorem \ref{['thm:CritExponents']}\ref{['thm:CritExponents - Susceptibility']}
• Definition 3.2