Random line tessellations of the plane: statistical properties of many-sided cells

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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≥ 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 pₙ(α) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of pₙ(α) up to terms that vanish as n→∞. 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 pₙ(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.