Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells

TL;DR

The article analyzes Poisson hyperplane percolation in hyperbolic space, establishing that the zero cell is almost surely bounded at the critical intensity for all dimensions $d\ge2$, and providing a complete phase-transition description: no unbounded cells above or at critical intensity and infinitely many unbounded cells below. It extends the planar result of Porret-Blanc to higher dimensions using a continuum Burton–Keane framework based on encounter points, the Mass Transport Principle, and hyperplane geometry, adapted to the hyperbolic setting. Furthermore, it reveals a richer phase structure with $d-1$ distinct transitions across dimensions, giving explicit formulas for when all $k$-faces are bounded. These results deepen understanding of continuum percolation in negatively curved spaces and inform stochastic geometry in hyperbolic environments.

Abstract

We show that tessellations of hyperbolic space by isometry-invariant Poisson processes of $(d-1)$-dimensional hyperplanes do not have an unbounded cell at the critical intensity. This extends a result by Porret-Blanc for the hyperbolic plane (C. R. Acad. Sci. Paris, Ser. I, Vol. 344 (2007)) to dimensions $d\ge3$. We also show that for intensities strictly below the critical intensity, infinitely many unbounded cells exist, while for intensities larger than or equal to the critical intensity, no unbounded cell exists. This completely describes the basic phase transition of this continuum percolation model. Our proof uses a method from discrete percolation theory which we adapt to the continuum and combine with specific computations for Poisson hyperplane processes.

Figures (2)

• Figure 1: Left: Sketch of the situation in dimension $d=2$. Pictured is a hyperplane $H(u,t)$ (cyan) and the caps that it 'cuts off' from $\partial B_{\mathbb{R}^d}(o,1)$ and $\partial B_{\mathbb{R}^d}(o,\tanh(r))$ respectively (orange). The opening angle $\alpha$ of the former satisfies $\cos(\alpha) = t$, while the opening angle $\beta$ of the latter satisfies $\cos(\beta) = t/\tanh(r)$. Note that $\tanh(r)$ needs to be larger than $1/\sqrt{2}$ in order for the orange caps to be disjoint from the black caps. In dimension $2$, the set $G$ consists of the four points $\{(\pm 1,0),(0,\pm 1)\}$ (magenta). Right: The set $G$ (magenta) in dimension $d=3$. In general, the set $G$ has dimension $d-2$.
• Figure 2: If $y^* \notin H$, then there exists a hyperplane $H'$ separating $y^*$ and $H$ (illustrated in the Klein model).

Theorems & Definitions (36)

• Theorem 1.1
• Corollary 1.2: Description of the phase transition
• Remark 1.3: Volume of the typical cell at criticality
• Remark 1.4: Situation in $\mathbb{H}^2$
• Theorem 2.1: Mass Transport Principle, cf. BS01
• proof
• Lemma 2.2
• proof
• Definition 3.1: Encounter point
• Lemma 3.2: Unbounded hyperplane tessellations admit encounter points