Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell

TL;DR

This work establishes sharp asymptotics for the tail of the maximal nucleus-to-vertex distance $\mathcal{D}$ in the typical Poisson-Voronoi cell. The authors reduce the tail probability to the first moment of a count of far-off pointy vertices and prove that pairwise contributions are negligible, yielding $\mathbb{P}(\mathcal{D}\ge t) \sim C_d (d\kappa_d)^d t^{d(d-1)} e^{-\kappa_d t^d}$ with an explicit constant $C_d$. The constant is interpreted as the mean volume of a random simplex under a geometric pointy-condition, and is given explicitly by $C_d=\frac{1}{2^{d-1}\sqrt{\pi}(d-1)!}\frac{\Gamma(d/2)^d}{\Gamma((d+1)/2)^{d-1}}$. The extremal index for the maximum over a large window is shown to be $\theta=1/(2d)$. The results extend to a parametric family with intensity $\|\,x\|^{\alpha}$, yielding corresponding tail laws with constants $C_{d,\alpha}$ and $K_{d,\alpha}$ expressed via Beta/Gamma functions, thereby broadening connections to Crofton/zero cells and isotropic hyperplane tessellations.

Abstract

We consider the typical Poisson-Voronoi cell in the Euclidean space R d and in particular the maximal distance D from a vertex of that cell to its nucleus. We provide a sharp asymptotics for the tail distribution of D. As a byproduct, we prove that the extremal index related to the sequence of such distances for all Voronoi cells included in a large box is equal to (2d) -1 . This confirms a conjecture formulated by Chenavier and Robert. The explicit constant appearing in the estimate of the tail probability of D is proved to be the mean volume of a random simplex formed by uniformly distributed points on the unit sphere conditioned on satisfying some spatial condition.

Figures (16)

• Figure 1: The left-hand side picture is a cluster of big cells in the $2$-dimensional case containing at least one with maximal nucleus-vertex distance $\geq 5$ obtained by emptying a ball of radius $5$ in the spirit of Aldous. In the center, a pointy vertex $c$ of $\mathcal{C}_0$ in the $3$-dimensional case and the points $\mathrm X_1, \mathrm X_2, \mathrm X_3$ that together with $0$ determine it. On the right-hand side, the shaded triangle is the set $\mathrm{Conv}(P_{\mathrm U_0^\perp}(\mathrm U_1), P_{\mathrm U_0^\perp}(\mathrm U_2), P_{\mathrm U_0^\perp}(\mathrm U_3))$ obtained from the same set of points: it contains the origin, showing that condition of Lemma \ref{['prop_equivalent_characterization_of_VOM']} is satisfied.
• Figure 2: (a) Shaded in red, the local shape of the Voronoi cell associated to $\mathrm{U}_0$ around a non pointy vertex generated by points $\mathrm U_0, \mathrm U_1 , \mathrm U_2, \mathrm U_3$. (b) In blue, the sets $B$ and $H$ together with the subspace $F$ and the unit vector $e$ in the context of the proof of Lemma \ref{['prop_equivalent_characterization_of_VOM']}.
• Figure 3: Representation of the classical Blaschke-Petkantschin spherical variable change $(\mathrm x_1,...,\mathrm x_d) \leftrightarrow (\mathrm r,\mathrm u_0,...,\mathrm u_d)$.
• Figure 4: Simulation of a cell with nucleus $\mathrm x$, shaded in gray, featuring a large maximal nucleus-vertex distance and two pointy vertices $c$ and $c'$ located at the center of respectively $\mathcal{B}$ and $\mathcal{B}'$. In this realization, $c$ and $c'$ share an edge, and therefore are determined by two common points of the Poisson point process: $\mathrm x$ and the point colored both in red and in blue.
• Figure 5: Sketch of the Blaschke-Petkantschin-type variable change of Lemma \ref{['prop_Nikitenko']} for the case $d=3,k=1$.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 2