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Visibility and intersection density for Boolean models in hyperbolic space

Tillmann Bühler, Daniel Hug, Christoph Thaele

TL;DR

This work extends stochastic-geometry analysis to hyperbolic space by defining and quantifying two Euclidean-proxy observables—the intersection density and the mean visible volume—for Poisson particle processes in $\mathbb{H}^d$. It develops hyperbolic intrinsic volumes via a Steiner-type expansion, connects them to Euclidean analogues, and derives an explicit, dimensionally uniform formula for the mean visible volume: $\overline{\mathrm{V}}_s(Z) = \frac{d!\kappa_d}{2^d} \frac{\Gamma((v_{d-1}^*\gamma - d + 1)/2)}{\Gamma((v_{d-1}^*\gamma + d + 1)/2)}$, with finiteness iff $v_{d-1}^*\gamma > d-1$ (equivalently $\gamma v_{d-1} > (d-1)\frac{d\kappa_d}{\kappa_{d-1}}$). The paper also provides an explicit formula for the intersection density and a detailed comparison with hyperbolic Poisson hyperplane tessellations, revealing a hyperbolic-threshold behavior absent in Euclidean space and establishing universal near-critical scaling laws. Together, these results deepen understanding of visibility and boundary interactions in negatively curved spaces and furnish exact tools for analyzing hyperbolic stochastic geometries.

Abstract

For Poisson particle processes in hyperbolic space we introduce and study concepts analogous to the intersection density and the mean visible volume, which were originally considered in the analysis of Boolean models in Euclidean space. In particular, we determine a necessary and sufficient condition for the finiteness of the mean visible volume of a Boolean model in terms of the intensity and the mean surface area of the typical grain.

Visibility and intersection density for Boolean models in hyperbolic space

TL;DR

This work extends stochastic-geometry analysis to hyperbolic space by defining and quantifying two Euclidean-proxy observables—the intersection density and the mean visible volume—for Poisson particle processes in . It develops hyperbolic intrinsic volumes via a Steiner-type expansion, connects them to Euclidean analogues, and derives an explicit, dimensionally uniform formula for the mean visible volume: , with finiteness iff (equivalently ). The paper also provides an explicit formula for the intersection density and a detailed comparison with hyperbolic Poisson hyperplane tessellations, revealing a hyperbolic-threshold behavior absent in Euclidean space and establishing universal near-critical scaling laws. Together, these results deepen understanding of visibility and boundary interactions in negatively curved spaces and furnish exact tools for analyzing hyperbolic stochastic geometries.

Abstract

For Poisson particle processes in hyperbolic space we introduce and study concepts analogous to the intersection density and the mean visible volume, which were originally considered in the analysis of Boolean models in Euclidean space. In particular, we determine a necessary and sufficient condition for the finiteness of the mean visible volume of a Boolean model in terms of the intensity and the mean surface area of the typical grain.
Paper Structure (7 sections, 12 theorems, 112 equations, 2 figures, 1 table)

This paper contains 7 sections, 12 theorems, 112 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic intrinsic volumes
  4. Stationary Poisson particle processes in hyperbolic space
  5. Intersection density of a hyperbolic Poisson particle process
  6. Mean visible volume in a hyperbolic Boolean model
  7. Comparison with hyperbolic Poisson hyperplane tessellations

Key Result

Proposition 3.1

Let $1 \leq k \leq d-1$ and identify $\mathbb{H}^k$ with an arbitrary $L_k \in A_h(d,k)$ via an isometry $\iota \colon \mathbb{H}^k \to \mathbb{H}^d$ such that $\iota(\mathbb{H}^k)=L_k$. If $A \in {\mathcal{K}}^k$, then

Figures (2)

  • Figure 1: Simulation of a Boolean model in the hyperbolic plane whose grains are discs of random radius uniformly distributed in $[0,1]$. The saturation of a point represents the number of grains by which it is covered.
  • Figure 2: Simulations of Boolean models with intensity $0.5$ (left), $0.9595$ (middle) and $1.5$ (right) in the hyperbolic plane whose grains are discs of radius $1/2$. The saturation of a point represents the number of grains by which it is covered. The point $\mathsf{p}$ is located at the center.

Theorems & Definitions (28)

  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • proof : Proof of \ref{['prop:intrinsic_hyp']}
  • Theorem 5.1
  • Remark 5.2
  • Lemma 5.3
  • ...and 18 more