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 to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson-Delaunay tessellation is also discussed.
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…
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.
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…
We make use of the recent proof that the critical probability for percolation on random Voronoi tessellations is 1/2 to prove the corresponding result for random Johnson-Mehl tessellations, as well as for two-dimensional slices of higher dimensional Voronoi tessellations. Surprisingly, the proof is a little simpler for these more complicated models.
The typical cell of a Voronoi tessellation generated by $n+1$ uniformly distributed random points on the $d$-dimensional unit sphere $\mathbb S^d$ is studied. Its $f$-vector is identified in distribution with the $f$-vector of a beta' polytope generated by $n$ random points in $\mathbb R^d$. Explicit formulae for the expected $f$-vector are provided for any $d$ and the low-dimensional cases $d\in\{2,3,4\}$ are studied separately. This implies an explicit formula for the total number of $k$-dimensional faces in the spherical Voronoi tessellation as well.
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…
We prove that the Palm measure of the Ginibre process is obtained by removing a Gaussian distributed point from the process and adding the origin. We obtain also precise formulas describing the law of the typical cell of Ginibre--Voronoi tessellation. We show that near the germs of the cells a more important part of the area is captured in the Ginibre--Voronoi tessellation than in the Poisson--Voronoi tessellation. Moment areas of corresponding subdomains of the cells are explicitly evaluated.
Position $n$ points uniformly at random in the unit square $S$, and consider the Voronoi tessellation of $S$ corresponding to the set $η$ of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let $H_S$ denote the event that there exists a red horizontal crossing of $S$ in the resulting colouring. In 1999, Benjamini, Kalai and Schramm conjectured that knowing the tessellation, but not the colouring, asymptotically gives no information as to whether the event $H_S$ will occur or not. More precisely, since $H_S$ occurs with probability …
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.
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.
Convergence towards Ideal Poisson--Voronoi tessellations with a focus on Diestel--Leader graphs
Institut Élie Cartan de Lorraine, CNRS, Université de Lorraine（洛林埃利嘉当研究所，法国国家科学研究中心，洛林大学）；School of Mathematics, Institute for Research in Fundamental Sciences（基础科学研究院数学学院）
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.
We introduce a new class of spatial-temporal point processes based on Voronoi tessellations. At each step of such a process, a point is chosen at random according to a distribution determined by the associated Voronoi cells. The point is then removed, and a new random point is added to the configuration. The dynamics are simple and intuitive and could be applied to modeling natural phenomena. We prove ergodicity of these processes under wide conditions.
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 $λ$ on $M$. We obtain asymptotic expansions up to the second order for the means of several characteristics of the Voronoi cell associated with $x_0$, including its volume and number of vertices. In each case, the first term of the estimate is equal to the mean characteristic in the Euclidean setting while the second term may contain a particular curvature of $M$ at $x_0$: the scalar curvature in the case of the mean nu…
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.
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…
Consider the distances $\tilde{R}_o$ and $R_o$ from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the $d$-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of $\tilde{R}_o$ and $R_o$. First, using the well-known relationship between the 0-cell and the typical cell, we show that the random variable $\tilde{R}_o$ is equivalent in distribution to the contact distance of the Poisson point process. Next, we derive a multi-integral expression for the exact distribution of $R_o$. Fu…
We observe a realization of a stationary generalized weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling technique to construct an unbiased estimator of the average value of this geometric characteristic. Under mild conditions on the weights of the cells, we establish variance asymptotics and the asymptotic normality of the unbiased estimator as the observation window tends to the whole space. Moreover, the weak consistency is shown for this estimator.
We introduce a new class of dynamic point process models with simple and intuitive dynamics that are based on the Voronoi tessellations generated by the processes. Under broad conditions, these processes prove to be ergodic and produce, on stabilisation, a wide range of clustering patterns. In the paper, we present results of simulation studies of three statistical measures (Thiel's redundancy, van Lieshout and Baddeley's $J$-function and the empirical distribution of the Voronoi nearest neighbours' numbers) for inference on these models from the clustering behaviour in the stationary regime. …
This paper establishes expectation and variance asymptotics for statistics of the Poisson--Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.
We establish a general Berry-Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre-Voronoi tessellations and of Poisson-Voronoi tessellations. Moreover, we apply the gene…
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…
A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: the Voronoi tessellation of a stationary Poisson point process is an example. We show that for an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. Using this equivalence, we show that the simple ra…
This article describes, in elementary terms, a generic approach to produce discrete random tilings and similar random structures by using point process theory. The standard Voronoi and Delone tilings can be constructed in this way. For this purpose, convex polytopes are replaced by their vertex sets. Three explicit constructions are given to illustrate the concept.
We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $πλ/3$ as $λ\to0.$ This answers a question of Benjamini and Schramm.
Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on $\mathbb R^2$ is the $ε\to 0$ limit of simple random…
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.
For a compact convex set $K$ and a Poisson point process $η$, the union of all Voronoi cells with a nucleus in $K$ is the Poisson-Voronoi approximation of $K$. Lower and upper bounds for the variance and a central limit theorem for the volume of the Poisson-Voronoi approximation are shown. The proofs make use of so called Wiener-Itô chaos expansions and the central limit theorem is based on a more abstract central limit theorem for Poisson functionals, which is also derived.
We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to $1/2$ as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm.
We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic…
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.
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…
