New Monte Carlo method for planar Poisson–Voronoi cells

By a new Monte Carlo algorithm, we evaluate the sidedness probability p n 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 → ∞. 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 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.