Many-sided Poisson–Voronoi cells with only Gabriel neighbors

Abstract

Let pnG be the probability for a planar Poisson–Voronoi cell to be n-sided and have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of log⁡pnG in the limit of large n. We determine several statistical properties of a many-sided cell obeying this ‘Gabriel condition.’ In particular, the cell perimeter, when parametrized as a function τ(θ) of the polar angle θ, behaves as a Brownian bridge on the interval 0⩽θ⩽2π . We point out similarities and differences with related problems in random geometry.