Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915690937581568 |
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| author | Bühler, Tillmann Gusakova, Anna Recke, Konstantin |
| author_facet | Bühler, Tillmann Gusakova, Anna Recke, Konstantin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19425 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells Bühler, Tillmann Gusakova, Anna Recke, Konstantin Probability 82B43, 60D05 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. |
| title | Critical Poisson hyperplane percolation in hyperbolic space has no unbounded cells |
| topic | Probability 82B43, 60D05 |
| url | https://arxiv.org/abs/2512.19425 |