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Estimating the critical threshold for stretched-out hyperbolic random connection models

Matthew Dickson

TL;DR

This work advances the understanding of critical thresholds for stretched-out random connection models on hyperbolic space by developing a lace-expansion framework compatible with hyperbolic geometry. It combines Mecke’s formula, operator-theoretic tools, and the spherical transform to obtain an Ornstein–Zernike equation for the two-point function and to extract asymptotics for the critical intensity $\lambda_c(L)$ as the edge-length scale $L$ grows. The authors provide explicit leading-term expansions for the Boolean disc and heat-kernel RCMs, showing model-dependent corrections that decay exponentially in $L$, and demonstrate that the spectral-radius analysis plays a central role in locating the threshold. The results illustrate qualitative differences from Euclidean lattices, particularly in how the $L^1\to L^1$ and $2\to 2$ norms diverge and how the spherical transform drives the asymptotics. Overall, the paper contributes a perturbative, geometry-aware toolkit for continuum percolation on hyperbolic spaces with concrete model applications and detailed asymptotic expansions.

Abstract

This paper examines the model-dependent asymptotic behaviour of the critical threshold intensity for stretched-out random connection models (RCMs) on hyperbolic spaces. The proof uses lace expansion arguments, but has notable qualitative differences to the Euclidean case in how it evaluates spectral radii. The result is applied to the Boolean disc RCM and a heat kernel RCM.

Estimating the critical threshold for stretched-out hyperbolic random connection models

TL;DR

This work advances the understanding of critical thresholds for stretched-out random connection models on hyperbolic space by developing a lace-expansion framework compatible with hyperbolic geometry. It combines Mecke’s formula, operator-theoretic tools, and the spherical transform to obtain an Ornstein–Zernike equation for the two-point function and to extract asymptotics for the critical intensity as the edge-length scale grows. The authors provide explicit leading-term expansions for the Boolean disc and heat-kernel RCMs, showing model-dependent corrections that decay exponentially in , and demonstrate that the spectral-radius analysis plays a central role in locating the threshold. The results illustrate qualitative differences from Euclidean lattices, particularly in how the and norms diverge and how the spherical transform drives the asymptotics. Overall, the paper contributes a perturbative, geometry-aware toolkit for continuum percolation on hyperbolic spaces with concrete model applications and detailed asymptotic expansions.

Abstract

This paper examines the model-dependent asymptotic behaviour of the critical threshold intensity for stretched-out random connection models (RCMs) on hyperbolic spaces. The proof uses lace expansion arguments, but has notable qualitative differences to the Euclidean case in how it evaluates spectral radii. The result is applied to the Boolean disc RCM and a heat kernel RCM.

Paper Structure

This paper contains 30 sections, 43 theorems, 250 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. The Model and the Results
  4. The Model
  5. Hyperbolic space.
  6. Random connection model.
  7. Critical Intensity.
  8. Results
  9. Background and Discussion
  10. Outline of the Paper and the Argument
  11. Preliminaries
  12. Mecke's Formula.
  13. Operators.
  14. Lace Expansion for the Critical Intensity
  15. Characterising the Percolation Threshold
  16. ...and 15 more sections

Key Result

Theorem 2.2

For the Boolean disc RCM, as $L\to\infty$ Therefore

Figures (4)

  • Figure 1: Simulations of the RCM on ${\mathbb{H}^2}$ (represented with the Poincaré ball model) with adjacency function $\varphi\left(r\right) = \mathds 1\left\{r < r_0\right\}$ where $r_0=2\mathrm{arsinh}\left(\frac{1}{2\sqrt{\pi}}\right)\approx0.557$ is the radius of the unit $\mu$-volume disc. The blue region is the union of hyperbolic radius $\frac{1}{2}r_0$ discs centred on each vertex.
  • Figure 2: Contours of $A\left(x,b\right)=-3-2,-1,0,1,2,3$ in the Poincaré disc model of ${\mathbb{H}^2}$.
  • Figure 3: Sketch and labelling of the cross-section of the region $B_L(o)\cap B_L(r)$.
  • Figure 4: Labelling of vertices, angles, and side lengths of the hyperbolic triangle $\Delta ABC$ used in Lemmas \ref{['lem:sinerule']} and \ref{['lem:areaoftriangles']}.

Theorems & Definitions (95)

  • Definition 2.1: Boolean disc RCM
  • Theorem 2.2
  • Definition 2.3: Heat Kernel RCM on ${\mathbb{H}^3}$
  • Theorem 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.0
  • Definition 3.1
  • Lemma 3.1
  • ...and 85 more