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Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform

Matthew Dickson

TL;DR

This work proves the existence of a non-uniqueness phase for infinite clusters in marked random connection models on hyperbolic space ${\mathbb{H}^d}$ (extended by marks in $\mathcal{E}$) in a high volume-scaling regime. It introduces an approach based on the spherical transform to diagonalize key operators and bound the $L^2\to L^2$ norms of the adjacency and two-point operators, while controlling the triangle diagram to access mean-field critical exponents. The authors establish that the susceptibility and percolation thresholds satisfy $\lambda_T(L) \le \lambda_c(L) \le \lambda_u(L)$ and, under favorable scaling (notably volume-linear), show $\lambda_c(L) < \lambda_u(L)$ for large $L$, thus a non-uniqueness regime. They further derive mean-field critical exponents in suitable regimes and apply the results to scaled Boolean disc models and weight-dependent hyperbolic RCMs, including non-perturbative cases with graphs that are not locally finite. The methodology hinges on a geometric-analytic blend: operator bounds, spherical-transform diagonalization, and hyperbolic geometry, enabling sharp control even in non-locally-finite settings and extending known non-uniqueness phenomena beyond amenable Euclidean settings.

Abstract

A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models on the $d$-dimensional hyperbolic space, ${\mathbb{H}^d}$, in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on ${\mathbb{H}^d}$ to diagonalize convolution by the adjacency function and the two-point function and bound their $L^2\to L^2$ operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic random connection models. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some random connection models whose resulting graphs are almost surely not locally finite.

Non-Uniqueness Phase in Hyperbolic Marked Random Connection Models using the Spherical Transform

TL;DR

This work proves the existence of a non-uniqueness phase for infinite clusters in marked random connection models on hyperbolic space (extended by marks in ) in a high volume-scaling regime. It introduces an approach based on the spherical transform to diagonalize key operators and bound the norms of the adjacency and two-point operators, while controlling the triangle diagram to access mean-field critical exponents. The authors establish that the susceptibility and percolation thresholds satisfy and, under favorable scaling (notably volume-linear), show for large , thus a non-uniqueness regime. They further derive mean-field critical exponents in suitable regimes and apply the results to scaled Boolean disc models and weight-dependent hyperbolic RCMs, including non-perturbative cases with graphs that are not locally finite. The methodology hinges on a geometric-analytic blend: operator bounds, spherical-transform diagonalization, and hyperbolic geometry, enabling sharp control even in non-locally-finite settings and extending known non-uniqueness phenomena beyond amenable Euclidean settings.

Abstract

A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models on the -dimensional hyperbolic space, , in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on to diagonalize convolution by the adjacency function and the two-point function and bound their operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic random connection models. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some random connection models whose resulting graphs are almost surely not locally finite.

Paper Structure

This paper contains 26 sections, 33 theorems, 218 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Results
  4. Presenting the Model and the Questions
  5. Hyperbolic space.
  6. Marked hyperbolic random connection model.
  7. Critical Behaviour.
  8. General Results
  9. Applications to Specific Models
  10. Scaled Boolean Disc Model on ${\mathbb{H}^d}$.
  11. Weight-dependent hyperbolic RCMs.
  12. Background
  13. Preliminaries
  14. Hyperbolic space.
  15. Connections and Augmentations.
  16. ...and 11 more sections

Key Result

Theorem 2.2

If Assumption assump:finitelymany or Assumption assump:specialscale holds, then for sufficiently large parameter $L$ we have $\lambda_{\mathrm{u}}(L) > \lambda_{\mathrm{c}}(L)$.

Figures (5)

  • Figure 1: Plot of $Q^\mathbb{B}_d\left(\varrho;0\right)$ for $d=2,3,4,5$ using MATLAB.
  • Figure 2: Construction of the length $\mathcal{L}_*\left(\theta\right)$ in the Poincaré disc model.
  • Figure 3: Sketch of the construction of the sets $\mathcal{V}_0$ and $\mathcal{V}_1$ in ${\mathbb{H}^2}$.
  • Figure 4: Sketches of the example scaling and adjacency functions in Example \ref{['expl:annulus']}.
  • Figure 5: Plot of the scaling function $\sigma_L$ in Example \ref{['expl:manyAnnulii']} for $R=0.25$, using MATLAB.

Theorems & Definitions (75)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Proposition 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 65 more