Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells

2025/12/22 by Tillmann Bühler, Bühler, Tillmann, Anna Gusakova +3
Mathematics · #60D05 #82B43 #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.2512.19425

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

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≥3. 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.

Citations

• Indistinguishability of unbounded components in the occupied and vacant sets of Boolean models on symmetric spaces
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• Boolean models in hyperbolic space
• Haagerup property and group-invariant percolation
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• Dimensions of random covering sets in Riemann manifolds
• Asymptotics of visibility in the hyperbolic plane
• Insertion and Deletion Tolerance of Point Processes
• Visibility to infinity in the hyperbolic plane, despite obstacles

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