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

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 . Along with it comes a nearly complete understanding of the structure of the large cell. We show that two independent biased random walks executed by the polar angle determine the trajectory of the cell perimeter. We find the limit distributions 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.