Many-sided Poisson-Voronoi cells with only Gabriel neighbors

Abstract

Let $p_n^G$ be the probability for a planar Poisson-Voronoi cell to be $n$-sided {\it 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 $\log p_n^G$ 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 $\tau(\theta)$ of the polar angle $\theta$, behaves as a Brownian bridge on the interval $0\le\theta\le 2\pi$. We point out similarities and differences with related problems in random geometry.