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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.02607 |
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| _version_ | 1866910983000162304 |
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| author | Irlbeck, Matthias Kabluchko, Zakhar Müller, Tobias |
| author_facet | Irlbeck, Matthias Kabluchko, Zakhar Müller, Tobias |
| contents | We study the typical cell of the Poisson-Voronoi tessellation. We show that when divided by the $d$-th root of the intensity parameter $λ$ of the Poisson process times the volume of the unit ball, the inradius, outradius, diameter and mean width of the typical cell converge in probability to the constants $1/2, 1, 2, 2$ respectively, as the dimension $d\to\infty$. We also show that the width of the typical cell, when rescaled in the same way, is bounded between $2\sqrt{5}/(2+\sqrt{5})-o_d(1)$ and $3/2+o_d(1)$, with probability $1-o_d(1)$. These results in particular imply that, with probability $1-o_d(1)$, the Hausdorff distance between the typical cell and any ball is at least of the order of the diameter of the typical cell.
In addition, we show that for all $k$ with $d-k\to\infty$, with probability $1-o_d(1)$, all faces of dimension $k$ have a diameter that is of a much smaller order than the diameter, inradius, etc., of the full typical cell. The same is true for ''almost all'' faces of dimension $d-k$ with $k$ fixed. And, we show that the number of such faces is $\left( (k+1)^{(k+1)/2} / k^{k/2} \pm o_d(1) \right)^d$ with probability $1-o_d(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the shape of the typical Poisson-Voronoi cell in high dimensions Irlbeck, Matthias Kabluchko, Zakhar Müller, Tobias Probability We study the typical cell of the Poisson-Voronoi tessellation. We show that when divided by the $d$-th root of the intensity parameter $λ$ of the Poisson process times the volume of the unit ball, the inradius, outradius, diameter and mean width of the typical cell converge in probability to the constants $1/2, 1, 2, 2$ respectively, as the dimension $d\to\infty$. We also show that the width of the typical cell, when rescaled in the same way, is bounded between $2\sqrt{5}/(2+\sqrt{5})-o_d(1)$ and $3/2+o_d(1)$, with probability $1-o_d(1)$. These results in particular imply that, with probability $1-o_d(1)$, the Hausdorff distance between the typical cell and any ball is at least of the order of the diameter of the typical cell. In addition, we show that for all $k$ with $d-k\to\infty$, with probability $1-o_d(1)$, all faces of dimension $k$ have a diameter that is of a much smaller order than the diameter, inradius, etc., of the full typical cell. The same is true for ''almost all'' faces of dimension $d-k$ with $k$ fixed. And, we show that the number of such faces is $\left( (k+1)^{(k+1)/2} / k^{k/2} \pm o_d(1) \right)^d$ with probability $1-o_d(1)$. |
| title | On the shape of the typical Poisson-Voronoi cell in high dimensions |
| topic | Probability |
| url | https://arxiv.org/abs/2506.02607 |