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

Abstract

We achieve a detailed understanding of then-sided planar Poisson–Voronoi cell in the limit of largen. Letpn be the probabilityfor a cell to have n sides. We construct the asymptotic expansion oflogpn up to terms that vanish as n → ∞. We obtain the statistics of the lengths of the perimeter segments and of the anglesbetween adjoining segments: to leading order as n → ∞, and after appropriate scaling, these become independentrandom variables whose laws we determine; and to next order in1/n they have nontrivial long range correlations whose expressions we provide. Then-sided cell tends towardsa circle of radius (n/4πλ)1/2, where λ is the cell density; hence Lewis’s law for the average areaAn ofthe n-sided cell behaves as An ≃ cn/λ with c = 1/4. For n → ∞ the cell perimeter, expressed as a functionR(ϕ) of thepolar angle ϕ, satisfies d2R/d ϕ2 = F(ϕ), where F is the known Gaussian noise; we deduce from it the probability law for the perimeter’s longwavelength deviations from circularity. Many other quantities related to the asymptotic cellshape become accessible to calculation.