AI 中文总结
研究三维及以上双曲空间上泊松 - 沃罗诺伊渗流恰有一个无界簇的阈值。此前相关唯一性阈值在一些情况下随强度趋于零而趋于零，本文将表明对于三维及以上双曲空间，该阈值下确界大于零，回答了Grebík和Recke的问题。
AI 中文摘要
我们研究了在\(d\geq3\)的\(d\)维双曲空间\(\mathbb{H}^d\)上泊松 - 沃罗诺伊渗流恰好存在一个无界簇的阈值。根据Grebík和Recke以及d'Achille等人的最新结果，对于在包括笛卡尔积\(\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}\)（其中\(k,d_1,\dots,d_k\geq2\)）的一族几何空间的环境空间上定义的泊松 - 沃罗诺伊渗流，随着基础泊松点过程的强度\(\lambda\)趋于零，这个“唯一性阈值”\(p_u(\lambda)\)趋于零。相比之下，对于双曲平面\(\mathbb{H}^2\)上的泊松 - 沃罗诺伊渗流，Benjamini和Schramm表明当\(\lambda\)趋于零时\(p_u(\lambda)\)趋于一，并且对于所有\(\lambda>0\)，\(p_u(\lambda)>1/2\)。D'Achille和Curien的一个未发表的论证表明，对于\(d\geq3\)的\(\mathbb{H}^d\)上的泊松 - 沃罗诺伊渗流，唯一性阈值满足对于所有\(\lambda>0\)，\(p_u(\lambda)\leq1/2\)。这里我们将表明对于\(d\geq3\)的\(\mathbb{H}^d\)上的泊松 - 沃罗诺伊渗流，\(\inf_{\lambda>0}p_u(\lambda)>0\)。这回答了Grebík和Recke的一个问题。
英文摘要
We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\mathbb{H}^d$ for $d\geq 3$. By recent results of Grebík and Recke and d'Achille et al., this "uniqueness threshold" $p_u(λ)$ 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 $\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}$ with $k,d_1,\dots,d_k\geq 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane $\mathbb{H}^2$, Benjamini and Schramm have shown that $p_u(λ)$ tends to one as $λ$ tends to zero, and $p_u(λ)>1/2$ for all $λ>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$, the uniqueness threshold satisfies $p_u(λ)\leq 1/2$ for all $λ>0$. Here we will show that $\inf_{λ>0}p_u(λ)>0$ for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$. This answers a question of Grebík and Recke.
