Vanishing uniqueness thresholds in Voronoi percolation on products
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| Format: | Preprint |
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2025
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| _version_ | 1866911292402434048 |
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| author | D'Achille, Matteo Grebík, Jan Khezeli, Ali Recke, Konstantin Wilkens, Amanda |
| author_facet | D'Achille, Matteo Grebík, Jan Khezeli, Ali Recke, Konstantin Wilkens, Amanda |
| contents | We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(λ)$ at small intensities $λ>0$ based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including $k$-fold graph products of $d$-regular trees for $k\ge2,d\ge3$ and products of hyperbolic spaces $\mathbb H_{d_1}\times \ldots \times \mathbb H_{d_k}$ for $k\ge2, d_i\ge2$, complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_23317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vanishing uniqueness thresholds in Voronoi percolation on products D'Achille, Matteo Grebík, Jan Khezeli, Ali Recke, Konstantin Wilkens, Amanda Probability Dynamical Systems Group Theory 82B43, 51F99, 60D05, 60G55 We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(λ)$ at small intensities $λ>0$ based on the unbounded borders phenomenon of their underlining ideal Poisson--Voronoi tessellation. We apply our method to several concrete examples in both the discrete and the continuum setting, including $k$-fold graph products of $d$-regular trees for $k\ge2,d\ge3$ and products of hyperbolic spaces $\mathbb H_{d_1}\times \ldots \times \mathbb H_{d_k}$ for $k\ge2, d_i\ge2$, complementing a recent result of the second and fourth author for symmetric spaces of connected higher rank semisimple real Lie groups with property (T). We also provide new examples of non-amenable Cayley graphs with the FIID sparse unique infinite cluster property, answering positively a recent question of Pete and Rokob. |
| title | Vanishing uniqueness thresholds in Voronoi percolation on products |
| topic | Probability Dynamical Systems Group Theory 82B43, 51F99, 60D05, 60G55 |
| url | https://arxiv.org/abs/2511.23317 |