We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640--656] for intersection points of motion-invariant Poisson line processes in $\mathbb{R}^2$. Our proof is based on Hoeffding's decomposition of $U$-statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the ``method of moments'' used in [Adv. in Appl. Probab. 30 (1998) 640--656] to treat the case $d=2$. Moreover, we extend our central limit theorem in several directions. First we consider $k$-flat processes induced by Poisson hyperplane processes in $\mathbb{R}^d$ for $0\le k\le d-1$. Second we derive (asymptotic) confidence intervals for the intensities of these $k$-flat processes and, third, we prove multivariate central limit theorems for the $d$-dimensional joint vectors of numbers of $k$-flats and their $k$-volumes, respectively, in an increasing spherical region.
We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to infinity.
We consider the typical cell of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. It is well known that the expected vertex number of the typical cell is independent of the directional distribution of the hyperplane process. We give sharp bounds for the variance of this vertex number, showing, in particular, that the maximum of the variance is attained if and only if the distribution of the process is rotation invariant with respect to a suitable scalar product. The employed representation of the second moment of the vertex number is a special case of formulas prov…
For a stationary Poisson hyperplane tessellation $X$ in ${\mathbb R}^d$, whose directional distribution satisfies some mild conditions (which hold in the isotropic case, for example), it was recently shown that with probability one every combinatorial type of a simple $d$-polytope is realized infinitely often by the polytopes of $X$. This result is strengthened here: with probability one, every such combinatorial type appears among the polytopes of $X$ not only infinitely often, but with positive density.
We study the problem of generating a hyperplane tessellation of an arbitrary set $T$ in $\mathbb{R}^n$, ensuring that the Euclidean distance between any two points corresponds to the fraction of hyperplanes separating them up to a pre-specified error $δ$. We focus on random gaussian tessellations with uniformly distributed shifts and derive sharp bounds on the number of hyperplanes $m$ that are required. Surprisingly, our lower estimates falsify the conjecture that $m\sim \ell_*^2(T)/δ^2$, where $\ell_*^2(T)$ is the gaussian width of $T$, is optimal.
This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables and point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations.
Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map that sends x in K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R^n embeds into the Hamm…
A Poisson line tessellation is observed within a window. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in the window in the limit as the window is scaled to infinity. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.
Since the seminal work by Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible, yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. We provide in this paper a fundamental link between typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations using martingale techniques and general theory of piecewise deterministic Markov processes (PDMPs). As applications, new mean values and new distributional res…
Stationary Poisson processes of lines in the plane are studied whose directional distributions are concentrated on $k \ge 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles' classical result for the isotropic case is obtained by an approximation argument.
In this paper we show existence of all exponential moments for the total edge length in a unit disk for a family of planar tessellations based on stationary point processes. Apart from classical such tessellations like the Poisson-Voronoi, Poisson-Delaunay and Poisson line tessellation, we also treat the Johnson-Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.
We consider the Voronoi tessellation based on a homogeneous Poisson point process in $\mathbf{R}^{d}$. For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us…
We show that a tessellation generated by a small number of random affine hyperplanes can be used to approximate Euclidean distances between any two points in an arbitrary bounded set $T$, where the random hyperplanes are generated by subgaussian or heavy-tailed normal vectors and uniformly distributed shifts. We derive quantitative bounds on the number of hyperplanes needed for constructing such tessellations in terms of natural metric complexity measures of $T$ and the desired approximation error. Our work extends significantly prior results in this direction, which were restricted to Gaussia…
We consider tessellations of the Euclidean $(d-1)$-sphere by $(d-2)$-dimensional great subspheres or, equivalently, tessellations of Euclidean $d$-space by hyperplanes through the origin; these we call conical tessellations. For random polyhedral cones defined as typical cones in a conical tessellation by random hyperplanes, and for random cones which are dual to these in distribution, we study expectations for a general class of geometric functionals. They include combinatorial quantities, such as face numbers, as well as, for example, conical intrinsic volumes. For isotropic conical tessella…
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 certain definite shape, which is determined by the directional distribution of the underlying hyperplane process. This result is extended here to typical $k$-faces of the tessellation, for $k\in\{2,...,d-1\}$. This requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction.
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 directional distribution, the quantity $n^{\frac{2}{d-1}}\sqrt[n]{\mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the o…
Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for very specific size functionals and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution $φ$ and with respect to an arbitrary size functional $Σ$. More precisely, we investigate the distribution of the typical cell $Z$, conditioned on the event $\{Σ(Z)<a\}$, w…
We study a geometric property related to spherical hyperplane tessellations in $\mathbb{R}^{d}$. We first consider a fixed $x$ on the Euclidean sphere and tessellations with $M \gg d$ hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of $d\log(d)\log(M)$ such that the radius of the cell containing $x$ induced by these hyperplanes is bounded above by, up to constants, $d\log(d)\log(M)/M$. We extend this result to hold for all …
A homogeneous Poisson-Voronoi tessellation of intensity $γ$ is observed in a convex body $W$. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in $W$. We prove that when $γ\rightarrow\infty$, these four quantities converge to Gumbel or Weibull distributions up to a rescaling…
In this paper, we construct a new family of random series defined on $\R^D$, indexed by one scaling parameter and two Hurst-like exponents. The model is close to Takagi-Knopp functions, save for the fact that the underlying partitions of $\R^D$ are not the usual dyadic meshes but random Voronoi tessellations generated by Poisson point processes. This approach leads us to a continuous function whose random graph is shown to be fractal with explicit and equal box and Hausdorff dimensions. The proof of this main result is based on several new distributional properties of the Poisson-Voronoi tesse…
Consider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $θ$. The first one is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and are such that the angle with which they see the two nuclei defining the facet is $θ$. The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight…
Typical weighted random simplices $Z_μ$, $μ\in(-2,\infty)$, in a Poisson-Delaunay tessellation in $\mathbb{R}^n$ are considered, where the weight is given by the $(μ+1)$st power of the volume. As special cases this includes the typical ($μ=-1$) and the usual volume-weighted ($μ=0$) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_μ$ satisfies a central limit theorem in high dimensions, that is, as $n\to\infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are st…
The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in $\mathbb{R}^d$ with $\ell$-dimensional affine subspaces, $1\leq \ell\leq d-1$, are shown to be random tessellations of the same type but with different model parameters. In particular, the intersection of a classical Poisson-Voronoi tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi tessellation. The geometric properties of the typical cell and, more generally, typical $k$-faces, of the sectional Poisson-Voronoi tessellation are studied in detail. It is…
We observe stationary random tessellations $X=\{Ξ_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $Ξ_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In …
In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson-Voronoi tessellation in the $d$-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition $Z$ of these two processes, two points of $Z$ are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting…
We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tes…
Given a stationary and isotropic Poisson hyperplane process and a convex body $K$ in ${\mathbb R}^d$, we consider the random polytope defined by the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. We investigate how well the expected mean width of this random polytope approximates the mean width of $K$ if the intensity of the hyperplane process tends to infinity.
The $β$-Delaunay tessellation in $\mathbb{R}^{d-1}$ is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a $β$-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in $\mathbb{R}^{d-1}$. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional $β$-Delaunay tessellation is analysed, as $d\to\infty$. In…
We consider a stationary Poisson hyperplane process with given directional distribution and intensity in $d$-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body $K$ and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing $K$. We study how well these random polytopes approximate $K$ (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of $K$.
We study the asymptotic behavior of a size-marked point process of centers of large cells in a stationary and isotropic Poisson hyperplane mosaic in dimension $d \ge 2$. The sizes of the cells are measured by their inradius or their $k$th intrinsic volume ($k \ge 2$), for example. We prove a Poisson limit theorem for this process in Kantorovich-Rubinstein distance and thereby generalize a result in Chenavier and Hemsley (2016) in various directions. Our proof is based on a general Poisson process approximation result that extends a theorem in Bobrowski, Schulte and Yogeshwaran (2021).
It is well known that the distributions of the interiors of the typical cell of a Poisson line tessellation and a STIT tessellation with the same parameters coincide. In this paper, differences in the arrangement of the cells in these two tessellation models are investigated. In particular, characteristics of the set of cells neighbouring the typical cell are studied. Furthermore, the pair-correlation function and several mark correlation functions of the point processes of cell centres are estimated and compared.
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 isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent…
There are various models of first passage percolation (FPP) in $\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\mathbb Z^d$ to $\mathbb R^d$ and adapt the results of \cite{boivin1990first} to prove a shape theorem for ergodic random pseudometrics on $\mathbb R^d$. A natural application of this result will be the study of FPP on random tessellations where a fluid starts in the zero cell and takes a random time to pass through the boundary of a cell into a neighbouring cell. We find that a tame rand…
Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. The purpose of this paper is to describe large scale asymptotic geometry of STIT tessellations in $\mathbb{R}^d$ and more generally that of non-stationary iteration infinitely divisible tessellations. We study several aspects of the typical first-order geometry of such tessellations resorting to martingale techniques…
An analogue of the classical Mecke formula for Poisson point processes is proved for the class of space-time STIT tessellation processes. From this key identity the Markov property of a class of associated random processes is derived. This in turn is used to determine the distribution of the number of internal vertices of the typical maximal tessellation segment.
For cell-division processes in a window, Cowan introduced four selection rules and two division rules each of which stands for one cell-division model. One of these is the area-weighted in-cell model. In this model, each cell is selected for division with a probability that corresponds to the ratio between its area and the area of the whole window. This selected cell is then divided by throwing a (uniformly distributed) point into the cell and drawing a line segment through the point under a random angle with the segment ending at the cell's boundary. For the STIT model which uses both a diffe…
This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$, where only hyperplanes that intersect a centred ball of radius $R>0$ are considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process converges in distribution, as $t\to\infty$, to a Poisson point process on $\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distan…
A Gilbert tessellation arises by letting linear segments (cracks) in the plane unfold in time with constant speed, starting from a homogeneous Poisson point process of germs in randomly chosen directions. Whenever a growing edge hits an already existing one, it stops growing in this direction. The resulting process tessellates the plane. The purpose of the present paper is to establish law of large numbers, variance asymptotics and a central limit theorem for geometric functionals of such tessellations. The main tool applied is the stabilization theory for geometric functionals.
We generalize the standard site percolation model on the $d$-dimensional lattice to a model on random tessellations of $\mathbb R^d$. We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument \cite{burton1989density}, develop two frameworks that imply the non-triviality of the phase transition and show that large classes of random tessellations fit into one of these frameworks. Our focus is on a very general approach that goes well beyond the typical Poisson driven models. The most interesting examples might be Voronoi tessellations induced by determinantal processe…
The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimensio…
Let $\mathfrak{m}$ be a random tessellation in $\mathbf{R}^d$ observed in a bounded Borel subset $W$ and $f(\cdot)$ be a measurable function defined on the set of convex bodies. To each cell $C$ of $\mathfrak{m}$ we associate a point $z(C)$ which is the nucleus of $C$. Applying $f(\cdot)$ to all the cells of $\mathfrak{m}$, we investigate the order statistics of $f(C)$ over all cells $C\in\mathfrak{m}$ with nucleus in $\mathbf{W}_ρ=ρ^{1/d}W$ when $ρ$ goes to infinity. Under a strong mixing property and a local condition on $\mathfrak{m}$ and $f(\cdot)$, we show a general theorem which reduces …
Slicing a Voronoi tessellation in $\mathbb{R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in $\mathbb{R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of…
