We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different "influence" parameters (called "speeds" in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex.
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 properties, such as the measure, of the Voronoi cells as $n$ tends to infinity. We adapt the methods used by Devroye et al (2017) to conduct a study of the asymptotic properties of two types of Voronoi cells: 1, Voronoi cells that have a fixed nucleus; 2, Voronoi cells that contain a fixed point. For the first type of Voronoi cells, we show that their geometr…
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.
For a Borel set $A$ and a homogeneous Poisson point process $η$ in $\R^d$ of intensity $λ>0$, define the Poisson--Voronoi approximation $ A_η$ of $A$ as a union of all Voronoi cells with nuclei from $η$ lying in $A$. If $A$ has a finite volume and perimeter we find an exact asymptotic of $\E\Vol(AΔA_η)$ as $λ\to\infty$ where $\Vol$ is the Lebesgue measure. Estimates for all moments of $\Vol(A_η)$ and $\Vol(AΔA_η)$ together with their asymptotics for large $λ$ are obtained as well.
In this paper, we are interested in the behavior of the typical Poisson-Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for its number of vertices and the area of the cell outside the disk. Moreover, for the latter, we establish a central limit theorem as well as moderate deviation type results. The proofs deeply rely on precise connections between Poisson-Voronoi tessellations, convex hulls of Poisson samples and germ-grain models in the unit ball. Besides, we derive analogous fac…
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 which is asymptotically described by two sequences of iid variables which are respectively Beta and exponentially distributed. This leads to a precise description of the scaling limit of a so-called typical cell. The limit object is a random apeirogon that we name menhir in reference to the Gallic huge stones. We also deduce from the aforementioned branching…
Consider a planar random point process made of the union of a point (the origin) and of a Poisson point process with a uniform intensity outside a deterministic set surrounding the origin. When the intensity goes to infinity, we show that the Voronoi cell associated with the origin converges from above to a deterministic convex set. We describe this set and give the asymptotics of the expectation of its defect area, defect perimeter and number of vertices. On the way, two intermediary questions are treated. First, we describe the mean characteristics of the Poisson-Voronoi cell conditioned on …
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 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…
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.
We study percolation in the following random environment: let $Z$ be a Poisson process of constant intensity in the plane, and form the Voronoi tessellation of the plane with respect to $Z$. Colour each Voronoi cell black with probability $p$, independently of the other cells. We show that the critical probability is 1/2. More precisely, if $p>1/2$ then the union of the black cells contains an infinite component with probability 1, while if $p<1/2$ then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kes…
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…
Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a measurable set. Construct the Voronoi cells of all points $x\in X$ with respect to $X$, and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in $K$. For $K$ a compact convex set the expectation of the volume difference $V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is computed. Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a Poisson point process. Concentration inequalities for both quantities ar…
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 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 …
