多面d维Poisson-Voronoi胞的启发式理论
Heuristic theory for many-faced d-dimensional Poisson-Voronoi cells
摘要：
我们认为d维Poisson-Voronoi曲面细分和调查的启发式方法最近开发的二维的适用性。设p_n（d）是一个细胞有n个邻居（n面）的概率，m_n（d）是与n面细胞相邻的细胞的平均面度。我们得到了p_n（d）和m_n（3）的渐近大n展开式的首阶项。看来，就像在二维，泊松-沃罗诺伊镶嵌违反的“线性定律”，也在三维。这一说法与现有的蒙特卡罗工作的对抗仍然是不确定的。然而，升级到当今计算机能力水平的模拟在原则上将能够证实（或无效）我们的理论。
英文摘要：
We consider the d-dimensional Poisson-Voronoi tessellation and investigate the applicability of heuristic methods developed recently for two dimensions. Let p_n(d) be the probability that a cell have n neighbors (be `n-faced') and m_n(d) the average facedness of a cell adjacent to an n-faced cell. We obtain the leading order terms of the asymptotic large-n expansions for p_n(d) and m_n(3). It appears that, just as in dimension two, the Poisson-Voronoi tessellation violates Aboav's `linear law' also in dimension three. A confrontation of this statement with existing Monte Carlo work remains inconclusive. However, simulations upgraded to the level of present-day computer capacity will in principle be able to confirm (or invalidate) our theory.
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