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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2503.24024 |
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| _version_ | 1866912301793148928 |
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| author | Bonnet, Gilles Gordon, Joseph |
| author_facet | Bonnet, Gilles Gordon, Joseph |
| contents | We investigate the typical cells $\widehat{Z}$ and $\widehat{Z}^\prime$ of $β$- and $β'$-Voronoi tessellations in $\mathbb{R}^d$, establishing a Complementary Theorem which entails: 1) a gamma distribution of the $Φ$-content (a suitable homogeneous functional) of the typical cell with $n$-facets; 2) the independence of this $Φ$-content with the shape of the cell; 3) a practical integral representation of the distribution of $Z^{(\prime)}$. We exploit the latter to derive bounds on the distribution of the facet numbers. Using duality, we get bounds on the typical degree distributions of $β$- and $β'$-Delaunay triangulations. For $β'$-Delaunay, the resulting exponential lower bound seems to be the first of its kind for random spatial graphs arising as the skeletons of random tessellations. For $β$-Delaunay, matching super-exponential bounds allow us to show concentration of the maximal degree in a growing window to only a finite number of deterministic values (in particular, only two values for $d=2$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degrees in the $β$- and $β'$-Delaunay graphs Bonnet, Gilles Gordon, Joseph Probability 52A22, 60D05, 60F05 (Priimary) 52B11, 60G55 (Secondary) We investigate the typical cells $\widehat{Z}$ and $\widehat{Z}^\prime$ of $β$- and $β'$-Voronoi tessellations in $\mathbb{R}^d$, establishing a Complementary Theorem which entails: 1) a gamma distribution of the $Φ$-content (a suitable homogeneous functional) of the typical cell with $n$-facets; 2) the independence of this $Φ$-content with the shape of the cell; 3) a practical integral representation of the distribution of $Z^{(\prime)}$. We exploit the latter to derive bounds on the distribution of the facet numbers. Using duality, we get bounds on the typical degree distributions of $β$- and $β'$-Delaunay triangulations. For $β'$-Delaunay, the resulting exponential lower bound seems to be the first of its kind for random spatial graphs arising as the skeletons of random tessellations. For $β$-Delaunay, matching super-exponential bounds allow us to show concentration of the maximal degree in a growing window to only a finite number of deterministic values (in particular, only two values for $d=2$). |
| title | Degrees in the $β$- and $β'$-Delaunay graphs |
| topic | Probability 52A22, 60D05, 60F05 (Priimary) 52B11, 60G55 (Secondary) |
| url | https://arxiv.org/abs/2503.24024 |