Asymptotic statistics of the n-sided planar Poisson–Voronoi cell: I. Exact results

We achieve a detailed understanding of the n-sided planar Poisson–Voronoi cell in the limit of large n. Let pn be the probability for a cell to have n sides. We construct the asymptotic expansion of logpn up to terms that vanish as n → ∞. We obtain the statistics of the lengths of the perimeter segments and of the angles between adjoining segments: to leading order as n → ∞, and after appropriate scaling, these become independent random variables whose laws we determine; and to next order in 1/n they have nontrivial long range correlations whose expressions we provide. The n-sided cell tends towards a circle of radius (n/4πλ)1/2, where λ is the cell density; hence Lewis's law for the average area An of the n-sided cell behaves as An cn/λ with c = 1/4. For n → ∞ the cell perimeter, expressed as a function R() of the polar angle , satisfies d2R/d 2 = F(), where F is the known Gaussian noise; we deduce from it the probability law for the perimeter's long wavelength deviations from circularity. Many other quantities related to the asymptotic cell shape become accessible to calculation.