双曲空间中水平球内的随机双曲多面体
Random hyperbolic polyhedra in horoballs
- TU Wien（维也纳工业大学）
- University of Münster（明斯特大学）
- Ruhr University Bochum（鲁尔大学波鸿分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究双曲空间中水平球内泊松点过程的测地凸包，将其边界投影为对偶泊松-拉盖尔镶嵌，证明临界状态下收敛于泊松-德劳内镶嵌，并给出体积、表面积及顶点强度的精确渐近结果。
AI中文摘要：
我们研究了$d$维双曲空间中限制在水平球内的平稳泊松点过程的测地凸包。由此产生的随机集合是一个无界双曲多面体，具有一个特殊的理想方向。将其边界小面投影到边界水平球面上，得到$\nmathbb{R}^{d-1}$上的平稳欧几里得镶嵌，我们将其识别为具有显式高度密度的对偶泊松-拉盖尔镶嵌。我们推导了其胞腔强度的精确公式，并且在临界状态下（即泊松点过程的强度与水平球的高度相匹配时），我们证明了投影镶嵌在$\nmathbb{R}^{d-1}$中局部收敛到经典的泊松-德劳内镶嵌。作为推论，典型胞腔依分布收敛，所有$k$维面的强度收敛到其泊松-德劳内对应值。我们还研究了双曲凸包的局部化体积泛函，确定了在同一状态下的极限期望，并使用埃夫隆型恒等式获得了顶点强度的二阶渐近展开。此外，我们推导了期望局部化表面积的精确公式，并确定了其临界渐近行为。在维度$d\ge3$时，期望的未归一化局部表面积收敛到有限极限，而在维度$d=2$时，它表现出对数增长。
英文摘要：
We study the geodesic convex hull of a stationary Poisson point process restricted to a horoball in $d$-dimensional hyperbolic space. The resulting random set is an unbounded hyperbolic polyhedron with a distinguished ideal direction. Projecting its boundary facets to the bounding horosphere yields a stationary Euclidean tessellation of $\mathbb{R}^{d-1}$, which we identify as a dual Poisson--Laguerre tessellation with an explicit height density. We derive an exact formula for its cell intensity and, in the critical regime where the intensity of the Poisson point process is matched with the height of the horoball, we prove local convergence of the projected tessellation to the classical Poisson--Delaunay tessellation in $\mathbb{R}^{d-1}$. As consequences, the typical cell converges in distribution and the intensities of all $k$-dimensional faces converge to their Poisson--Delaunay counterparts. We also study a localised volume functional of the hyperbolic convex hull, determine its limiting expectation in the same regime, and use an Efron-type identity to obtain a second-order asymptotic expansion for the vertex intensity. In addition, we derive an exact formula for the expected localised surface area and determine its critical asymptotics. In dimensions $d\ge3$ the expected unnormalised local surface area converges to a finite limit, whereas in dimension $d=2$ it exhibits logarithmic growth.
