Random 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 \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.
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Submitted on : Friday, February 22, 2008 - 4:56:34 PM
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  • HAL Id : hal-00258616, version 1
  • ARXIV : 0802.1869



H. J. Hilhorst, Pierre Calka. Random tessellations of the plane: statistical properties of many-sided cells. Journal of Statistical Physics, Springer Verlag, 2008, 132, pp.627-647. ⟨hal-00258616⟩



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