New Monte Carlo method for planar Poisson-Voronoi cells

By a new Monte Carlo algorithm we evaluate the sidedness probability pn of a planar Poisson-Voronoi cell in the range 3 ≤ n ≤ 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of pn as n → ∞. Our pn 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 n−1. 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.