典型泊松 - 沃罗诺伊胞体体积的精确尺度 - 形状分解
Exact Scale--Shape Factorization of the Typical Poisson--Voronoi Cell Volume in Arbitrary Dimension
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中文总结 AI 辅助
研究欧几里得空间中平稳泊松 - 沃罗诺伊镶嵌的典型胞体体积，通过推导精确尺度 - 形状分解，得出沃罗诺伊花体积性质，得到相关恒等式和界，简化下尾问题并推导渐近，模拟验证结果。
中文摘要 AI 辅助
对于欧几里得空间中的平稳泊松 - 沃罗诺伊镶嵌，我们推导了帕尔姆典型胞体体积的精确尺度 - 形状分解。在有效面数的条件下，沃罗诺伊花体积服从伽马分布且与归一化形状无关。这产生了精确的混合表示、变换和矩恒等式，以及归一化胞体与花比率的通用界。还将形状变异性在固定面数时与面数混合分离为偏离单一伽马定律的两个来源。我们将下尾问题简化为关键的逆体积积分和归一化配置空间上单独的高面可和性条件，并推导了条件主导的小体积渐近。一维情况可精确恢复，二维情况有明确坐标，二维到四维的模拟说明了主要结果。
英文摘要
Despite more than six decades of research, a tractable closed-form distribution for the typical Poisson--Voronoi cell volume remains unknown beyond one dimension. Building on the classical complementary-theorem structure for the Poisson--Voronoi fundamental region, we develop an explicit configuration-space factorization of the Palm-typical cell volume for a tessellation generated by a stationary Poisson point process of intensity \(λ>0\) in \(\mathbb R^d\), \(d\geq1\). Conditional on the number \(k\) of effective facets, the Voronoi flower volume is a \(\operatorname{Gamma}(k,\operatorname{rate}=λ)\) scale variable independent of the effective-neighbour configuration normalized to have unit flower volume. Mapping this normalized configuration to its cell-to-flower volume ratio \(A_k\) gives the conditional cell-volume representation as the product of the classical Gamma scale and a bounded geometric shape factor. From this representation, we derive exact mixture formulae, transform and moment identities, and criteria characterizing when the conditional cell-volume laws are Gamma. We also distinguish shape-factor variability within facet-number strata from mixing across strata as two sources of departure from a single Gamma law and recover the universal bound \(0<A_k\leq2^{-d}\). For the lower tail, we reduce critical negative moments of the shape factor to inverse-volume integrals over normalized configuration spaces. Under explicit critical-integrability and higher-facet summability assumptions, the density of the intensity-normalized cell volume has leading order \(y^d\) as \(y\downarrow0\). The framework recovers the one-dimensional distribution, admits explicit planar coordinates, and supports numerical evaluation of the mixture representation in dimensions two through four.
