Limits of Poisson-Laguerre tessellations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Gusakova, Anna Wolde-Lübke, Mathias in |
| author_facet | Gusakova, Anna Wolde-Lübke, Mathias in |
| contents | For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(η_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $η_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n)_{n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $β$-Voronoi and the $β$-Delaunay tessellations, respectively, as $β\to -1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limits of Poisson-Laguerre tessellations Gusakova, Anna Wolde-Lübke, Mathias in Probability 60D05, 60G55, 60B10 For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(η_n)_{n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $η_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n)_{n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $β$-Voronoi and the $β$-Delaunay tessellations, respectively, as $β\to -1$. |
| title | Limits of Poisson-Laguerre tessellations |
| topic | Probability 60D05, 60G55, 60B10 |
| url | https://arxiv.org/abs/2602.06906 |