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

TL;DR

This paper analyzes the statistical properties of many-sided cells in a family of random line tessellations of the plane, revealing asymptotic behaviors and differences at a critical parameter value.

Contribution

It provides the asymptotic expansion of the probability for a zero-cell to have n sides across different parameter values, including the critical case of lpha=1.

Findings

Cells become circular as npproaches infinity

The zero-cell's position is delocalized at lpha=1

Asymptotic expansion differs at the critical point lpha=1

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 \alpha\geq 1. For \alpha=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for \alpha=2 it coincides with the typical Poisson-Voronoi cell. Let p_n(\alpha) 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(\alpha) 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 \alpha>1, but gets delocalized for the Crofton cell, \alpha=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.