High-intensity Voronoi percolation on manifolds
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912522524688384 |
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| author | Bühler, Tillmann Dembin, Barbara Radhakrishnan, Ritvik Ramanan Severo, Franco |
| author_facet | Bühler, Tillmann Dembin, Barbara Radhakrishnan, Ritvik Ramanan Severo, Franco |
| contents | We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $λ$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,λ)$ and $p_u(M,λ)$ converge to the Euclidean critical parameter $p_c(\mathbb{R}^d)$. This extends a recent result of Hansen & Müller in the special case $M=\mathbb{H}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\varepsilon$-nets, allowing us to implement a "fine-graining" argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21737 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | High-intensity Voronoi percolation on manifolds Bühler, Tillmann Dembin, Barbara Radhakrishnan, Ritvik Ramanan Severo, Franco Probability 82B43, 60K35, 60D05 We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $λ$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,λ)$ and $p_u(M,λ)$ converge to the Euclidean critical parameter $p_c(\mathbb{R}^d)$. This extends a recent result of Hansen & Müller in the special case $M=\mathbb{H}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\varepsilon$-nets, allowing us to implement a "fine-graining" argument. |
| title | High-intensity Voronoi percolation on manifolds |
| topic | Probability 82B43, 60K35, 60D05 |
| url | https://arxiv.org/abs/2503.21737 |