平面的随机线镶嵌：多边形单元的统计特性
Random line tessellations of the plane: statistical properties of many-sided cells
摘要：
我们考虑一个家庭的随机线镶嵌的欧几里德平面介绍了一个更正式的背景下，由拥抱和施耐德[几何。Anal. 17，156（2007）]，并由参数α \geq 1描述。当α=1时，零胞（即包含原点的胞）与泊松线镶嵌的克罗夫顿胞重合，当α=2时，它与典型的泊松-沃罗诺伊胞重合。设pn（α）为零胞有n条边的概率。本文用统计力学的方法构造了\log p_n（α）的渐近展开式，直到n个\to \infty消失为止.在大n极限下，细胞被证明是圆形的。当α>1时，圆以原点为中心，但对于Crofton胞，α=1，这是参数范围的奇点。因此，\log p_n（1）的大n展开式与一般情况下的大n展开式不同，我们给出了如何实现它。作为推论，我们得到了类似的扩展为{ \it典型} n边细胞的泊松线镶嵌。
英文摘要：
We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter α\geq 1. For α=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for α=2 it coincides with the typical Poisson-Voronoi cell. Let p_n(α) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of \log p_n(α) up to terms that vanish as n\to\infty. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when α>1, but gets delocalized for the Crofton cell, α=1, which is a singular point of the parameter range. The large-n expansion of \log p_n(1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the {\it typical} n-sided cell of a Poisson line tessellation.
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