Asymptotic statistics of then-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 p n be the probability for a cell to have n sides. We construct the asymptotic expansion of log p n 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 A n of the n -sided cell behaves as A n ≃ cn /λ with c = 1/4. For n → ∞ the cell perimeter, expressed as a function R (ϕ) of the polar angle ϕ, satisfies d 2 R /d ϕ 2 = F (ϕ), 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.