2026/07/20 by Matthias Irlbeck, Tobias Müller
#math.PR
We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the d-dimensional hyperbolic space ℍd for d≥ 3. By recent results of Grebík and Recke and d'Achille et al., this "uniqueness threshold" pu(λ) tends to zero as the intensity λ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products ℍd1×…×ℍdk with k,d1,…,dk≥ 2. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane ℍ2, Benjamini and Schramm have shown that pu(λ) tends to one as λ tends to zero, and pu(λ)>1/2 for all λ>0. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on ℍd with d≥ 3, the uniqueness threshold satisfies pu(λ)≤ 1/2 for all λ>0. Here we will show that infλ>0pu(λ)>0 for Poisson-Voronoi percolation on ℍd with d≥ 3. This answers a question of Grebík and Recke.