Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

We consider a family of random line tessellations of the Euclidean plane introduced in a more formal context by Hug and Schneider (Geom. Funct. Anal. 17:156, 2007) and described by a parameter α≥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. We construct the asymptotic expansion of log p _n (α) up to terms that vanish as n→∞. Our methods are nonrigorous but of the kind commonly accepted in theoretical physics as leading to exact results. 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 typical n-sided cell of a Poisson line tessellation.