Many-sided Poisson-Voronoi cells with only Gabriel neighbors

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.