AI 中文总结
本文研究$d$维单位球面上两种随机镶嵌模型，推导泊松超球面镶嵌典型边长及球面分裂镶嵌典型最大线段的长度分布，并计算二维情形下内部关联概率。
AI 中文摘要
我们研究了$d$维单位球面上两种随机镶嵌模型的分布性质。首先，我们分析了由超球面空间上的泊松过程生成的泊松超球面镶嵌。我们推导了其典型边长长度分布的显式公式。其次，我们转向球面分裂镶嵌，这是一类由几何依赖的马尔可夫分裂动力学驱动的自然随机镶嵌类。我们获得了典型最大线段长度分布的精确表达式。与泊松模型不同，这些最大线段可能表现出内部关联。对于$d=2$，我们显式计算了典型最大线段具有给定数量的此类内部关联的概率。我们的一些结果依赖于一个适用于球面分裂过程的新Mecke型公式。
英文摘要
We investigate distributional properties of two random tessellation models on the $d$-dimen\-sional unit sphere. First, we analyze the Poisson hypersphere tessellation generated by a Poisson process on the space of hyperspheres. We derive an explicit formula for the length distribution of its typical edge. Second, we turn to spherical splitting tessellations, which form a natural class of random tessellations driven by a geometry-dependent Markovian split dynamics. We obtain an exact expression for the length distribution of the typical maximal segment. Unlike in the Poisson model, these maximal segments may exhibit internal incidences. For \(d=2\), we explicitly compute the probability that the typical maximal segment has a given number of such interior incidences. Some of our results rely on a new Mecke-type formula adapted to the spherical splitting process.
