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arXiv · 2511.23317

Vanishing uniqueness thresholds in Voronoi percolation on products

Matteo D'Achille·Jan Grebík·Ali Khezeli·Konstantin Recke·Amanda Wilkens
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Abstract

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.

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Matteo D'Achille, Jan Grebík, Ali Khezeli, Konstantin Recke, Amanda Wilkens. 2025-11-28. Vanishing uniqueness thresholds in Voronoi percolation on products. https://arxiv.org/abs/2511.23317

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