Related papers: Many-faced cells and many-edged faces in 3D Poisso…

We consider the 3D Poisson-Voronoi tessellation. We investigate the joint probability distribution pi_n(L) for an arbitrarily selected cell face to be n-edged and for the distance between the seeds of its adjacent cells to be equal to 2L.…

We consider the d-dimensional Poisson-Voronoi tessellation and investigate the applicability of heuristic methods developed recently for two dimensions. Let p_n(d) be the probability that a cell have n neighbors (be `n-faced') and m_n(d)…

We consider the three-dimensional Poisson-Voronoi tessellation and study the average facedness m_n of a cell known to neighbor an n-faced cell. Whereas Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic…

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the…

We achieve a detailed understanding of the $n$-sided planar Poisson-Voronoi cell in the limit of large $n$. Let ${p}\_n$ be the probability for a cell to have $n$ sides. We construct the asymptotic expansion of $\log {p}\_n$ up to terms…

We develop a set of heuristic arguments to explain several results on planar Poisson-Voronoi tessellations that were derived earlier at the cost of considerable mathematical effort. The results concern Voronoi cells having a large number n…

We study the typical cell of the Poisson-Voronoi tessellation. We show that when divided by the $d$-th root of the intensity parameter $\lambda$ of the Poisson process times the volume of the unit ball, the inradius, outradius, diameter and…

Let $\mathcal Z_d$ be the zero cell of a $d$-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of $k$-dimensional faces of $\mathcal Z_d$, as $d\to\infty$. For…

We consider the Voronoi diagram generated by $n$ i.i.d. $\mathbb{R}^{d}$-valued random variables with an arbitrary underlying probability density function $f$ on $\mathbb{R}^{d}$, and analyse the asymptotic behaviours of certain geometric…

In this paper, we consider a Riemannian manifold $M$ and the Poisson-Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda$ on $M$. We obtain asymptotic expansions up to the…

Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $\mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the…

Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250,000,000 cells to provide…

We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter \alpha\geq 1. For \alpha=1 the zero-cell…

The Voronoi tessellation of a homogeneous Poisson point process in the lower half-plane gives rise to a family of vertical elongated cells in the upper half-plane. The set of edges of these cells is ruled by a Markovian branching mechanism…

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and…

We bridge the properties of the regular square and honeycomb Voronoi tessellations of the plane to those of the Poisson-Voronoi case, thus analyzing in a common framework symmetry-break processes and the approach to uniformly random…

Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however so far only with flat hypersurfaces as cell-cell contact borders. In order to reproduce the experimentally…

A generalized version of a well-known problem of D. G. Kendall states that the zero cell of a stationary Poisson hyperplane tessellation in ${\mathbb{R}}^d$, under the condition that it has large volume, approximates with high probability a…

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…

Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices. Although this particular space tessellation is intensively studied by mathematicians, in two- and three dimensional…