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 behaviour of p(n) as n -> 8. 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 behaviour of these quantities is analysed in terms of an 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.