三维Poisson-Voronoi胞的大n条件面性m_n
Large-n conditional facedness m_n of 3D Poisson-Voronoi cells
摘要：
我们考虑了三维Poisson-Voronoi曲面细分，并研究了与n面胞相邻的胞的平均面度m_n。鉴于帕夫定律规定m_n=A+B/n，理论论证表明渐近展开式m_n = 8 + k_1 n^{-1/6} +.。最近新的蒙特卡罗数据，由于拉扎尔等人，基于一个非常大的数据集，现在很明显排除了Mrsavav定律。在这项工作中，我们确定了k_1的数值，并比较的Monte Carlo数据的扩展。k_1的计算涉及到一个由圆弧组成的辅助平面细胞结构，我们称之为泊松-莫比乌斯图。它是更一般的莫比乌斯图（或乘法加权幂图）的一个特例，它本身就很有趣。我们得到了单位面积总边长的精确结果，这是系数k_1的先决条件，以及该图中的其他一些量。
英文摘要：
We consider the three-dimensional Poisson-Voronoi tessellation and study the average facedness m_n of a cell known to neighbor an n-faced cell. Whereas Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar et al., based on a very large data set, now clearly rule out Aboav's law. In this work we determine the numerical value of k_1 and compare the expansion to the Monte Carlo data. The calculation of k_1 involves an auxiliary planar cellular structure composed of circular arcs, that we will call the Poisson-Moebius diagram. It is a special case of more general Moebius diagrams (or multiplicatively weighted power diagrams) and is of interest for its own sake. We obtain exact results for the total edge length per unit area, which is a prerequisite for the coefficient k_1, and a few other quantities in this diagram.
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