Many-sided Poisson-Voronoi cells with only Gabriel neighbors
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917611688689664 |
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| author | Hilhorst, H. J. |
| author_facet | Hilhorst, H. J. |
| contents | Let $p_n^G$ be the probability for a planar Poisson-Voronoi cell to be $n$-sided {\it and\,} have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of $\log p_n^G$ in the limit of large $n$. We determine several statistical properties of a many-sided cell obeying this `Gabriel condition.' In particular, the cell perimeter, when parametrized as a function $τ(θ)$ of the polar angle $θ$, behaves as a Brownian bridge on the interval $0\leθ\le 2π$. We point out similarities and differences with related problems in random geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Many-sided Poisson-Voronoi cells with only Gabriel neighbors Hilhorst, H. J. Statistical Mechanics Probability Let $p_n^G$ be the probability for a planar Poisson-Voronoi cell to be $n$-sided {\it and\,} have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of $\log p_n^G$ in the limit of large $n$. We determine several statistical properties of a many-sided cell obeying this `Gabriel condition.' In particular, the cell perimeter, when parametrized as a function $τ(θ)$ of the polar angle $θ$, behaves as a Brownian bridge on the interval $0\leθ\le 2π$. We point out similarities and differences with related problems in random geometry. |
| title | Many-sided Poisson-Voronoi cells with only Gabriel neighbors |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2403.07677 |