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

Abstract

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.

26 pages, 3 figures

References in corpus (5)

• Limit theorems for the typical Poisson-Voronoi cell and the Crofton cell with a large inradius
• Planar Voronoi cells : the violation of Aboav's law explained
• New Monte Carlo method for planar Poisson-Voronoi cells
• Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results
• The perimeter of large planar Voronoi cells: a double-stranded random walk

Cited by in corpus (4)

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• Asymptotic statistics of the n-sided planar Voronoi cell: II. Heuristics
• Sylvester's question and the Random Acceleration Process
• Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation