The perimeter of large planar Voronoi cells: a double-stranded random walk

Details The perimeter of large planar Voronoi cells: a double-stranded random walk The perimeter of large planar Voronoi cells: a double-stranded random walk

2004-12-21

arxiv.org/abs/cond-mat/0412569v2

Journal of Statistical Mechanics: Theory and Experiment -- (2005) L02003

Physics

Condensed Matter

Statistical Mechanics

Slightly extended version; journal reference added

Scientific paper

10.1088/1742-5468/2005/02/L02003

Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to terms that vanish as $n\to\infty$. We show that {\it two independent biased random walks} executed by the polar angle determine the trajectory of the cell perimeter. We find the limit distribution of (i) the angle between two successive vertex vectors, and (ii) the one between two successive perimeter segments. We obtain the probability law for the perimeter's long wavelength deviations from circularity. We prove Lewis' law and show that it has coefficient 1/4.

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