New Monte Carlo method for planar Poisson-Voronoi cells

Abstract

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have between four and six significant digits. Accurate n dependent averages, second moments, and variances are obtained for the cell area and the cell perimeter. The numerical large n behavior of these quantities is analyzed in terms of asymptotic power series in 1/n. Snapshots are shown of typical occurrences of extremely rare events implicating cells of up to n=1600 sides embedded in an ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.

35 pages including 10 figures and 4 tables

References in corpus (2)

• 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

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• Sylvester's question and the Random Acceleration Process
• Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation
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