The volume of hyperbolic Poisson zero cells: critical divergence and exact second moment
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arXiv
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| Format: | Preprint |
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2026
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| author | Bühler, Tillmann Thäle, Christoph |
| author_facet | Bühler, Tillmann Thäle, Christoph |
| contents | We investigate the second volume moment of the zero cell $Z_o$ of a Poisson hyperplane tessellation with intensity $γ$ in the $d$-dimensional hyperbolic space. We focus on the phase transition at the critical intensity $γ_c^{(d)}$, the minimum value for which $Z_o$ is almost surely bounded. In the critical regime $γ=γ_c^{(d)}$, we show that the second volume moment of the restricted zero cell $Z_o \cap B_R$, where $B_R$ is a hyperbolic ball of radius $R$ centred at $o$, diverges in any dimension at the universal rate $R^3$ as $R \to \infty$. In the supercritical case $γ> γ_c^{(d)}$, we prove that the full second volume moment is finite. Using tools from harmonic analysis in hyperbolic space, we derive an exact expression for this moment in terms of the Meijer $G$-function. Furthermore, we determine the asymptotic behaviour of the second moment as $γ\to \infty$ and as $γ\downarrow γ_c^{(d)}$, facilitating a direct comparison with the corresponding Euclidean values as well as the mean-field universality class of percolation theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05760 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The volume of hyperbolic Poisson zero cells: critical divergence and exact second moment Bühler, Tillmann Thäle, Christoph Probability 60D05 (Primary) 33C60, 43A85, 51M10, 52A22 (Secondary) We investigate the second volume moment of the zero cell $Z_o$ of a Poisson hyperplane tessellation with intensity $γ$ in the $d$-dimensional hyperbolic space. We focus on the phase transition at the critical intensity $γ_c^{(d)}$, the minimum value for which $Z_o$ is almost surely bounded. In the critical regime $γ=γ_c^{(d)}$, we show that the second volume moment of the restricted zero cell $Z_o \cap B_R$, where $B_R$ is a hyperbolic ball of radius $R$ centred at $o$, diverges in any dimension at the universal rate $R^3$ as $R \to \infty$. In the supercritical case $γ> γ_c^{(d)}$, we prove that the full second volume moment is finite. Using tools from harmonic analysis in hyperbolic space, we derive an exact expression for this moment in terms of the Meijer $G$-function. Furthermore, we determine the asymptotic behaviour of the second moment as $γ\to \infty$ and as $γ\downarrow γ_c^{(d)}$, facilitating a direct comparison with the corresponding Euclidean values as well as the mean-field universality class of percolation theory. |
| title | The volume of hyperbolic Poisson zero cells: critical divergence and exact second moment |
| topic | Probability 60D05 (Primary) 33C60, 43A85, 51M10, 52A22 (Secondary) |
| url | https://arxiv.org/abs/2604.05760 |