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 $τ(θ)$ of the polar angle $θ$, behaves as a Brownian bridge on the interval $0\leθ\le 2π$. We point out similarities and differences with related problems in random geometry.

Figures (1)

• Figure 1: Heavy black line: the perimeter of a six-sided Poisson-Voronoi cell having its seed in the origin ${\bf O}$. The seeds of the neighboring cells, not shown, are at positions $2{\bf{R}}_m$, where $m=1,\ldots,n$. Thin solid black lines and thin dashed red lines connect the origin to the midpoints ${\bf{R}}_m$ and the vertices ${\bf{S}}_m$, respectively The figure defines the angles $\beta_m$ as the increments of the polar angle when one advances along the perimeter from ${\bf{R}}_{m-1}$ to ${\bf{S}}_m$, and the $\gamma_m$ as the increments when one advances from ${\bf{S}}_m$ to ${\bf{R}}_m$. The $m$th neighbor cell is a Gabriel neighbor if ${\bf{R}}_m$ belongs to the perimeter segment $\overline{{{\bf{S}}_m{\bf{S}}_{m+1}}}$, and is not if ${\bf{R}}_m$ belongs to its extension. The latter situation occurs for $m=4$, with ${\bf{R}}_4$ located on the extension of segment $\overline{{{\bf{S}}_4{\bf{S}}_5}}$, outside of the segment itself. Hence this perimeter segment is not Gabriel; it is associated with a negative angle $\gamma_4$. For a general Poisson-Voronoi cell having its seed in ${\bf O}$ the $\beta_m$ and $\gamma_m$ may be of either sign; but it is easily seen that the cell has only Gabriel neighbors if and only if the $\beta_m$ and $\gamma_m$ are positive for all $m$.