Pierre Calka

The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane. ∗

Among the disks centered at a typical particle of the two-dimensional Poisson-Voronoi tessellation, let R m be the radius of the largest included within the polygonal cell associated with that particle and R M be the radius of the smallest containing that polygonal cell. In this article, we obtain the joint distribution of R m and R M. This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The same method works for studying the Crofton cell associated with the Poisson line process in the plane. The computation of the conditional probabilities P{R M ≥ r + s | R m = r} reveals the circular property of the Poisson-Voronoi typical cells (as well as the Crofton cells) having a ‘large’ in-disk.

Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process

In this paper, we give an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell 𝒞 of a two-dimensional Poisson-Voronoi tessellation. We deduce from it precise formulae for the distributions of the principal geometric characteristics of 𝒞 (area, perimeter, area of the fundamental domain). We also adapt the method to the Crofton cell and the empirical (or typical) cell of a Poisson line process.

Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball Bd, d≥2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.

LIMIT THEOREMS FOR THE TYPICAL POISSON-VORONOI CELL AND THE CROFTON CELL WITH A LARGE INRADIUS

In this paper, we are interested in the behaviour 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 innit y. 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 facts for the Crofton cell of a stationary Poisson line process in the plane.

AN EXPLICIT EXPRESSION FOR THE DISTRIBUTION OF THE NUMBER OF SIDES OF THE TYPICAL POISSON-VORONOI CELL

In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.

Large deviation probabilities for the number of vertices of random polytopes in the ball

In this paper we establish large deviation results on the number of extreme points of a homogeneous Poisson point process in the unit ball of R d . In particular, we deduce an almost-sure law of large numbers in any dimension. As an auxiliary result we prove strong localization of the extreme points in an annulus near the boundary of the ball.

Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

We consider a family of random line tessellations of the Euclidean plane introduced in a more formal context by Hug and Schneider (Geom. Funct. Anal. 17:156, 2007) and described by a parameter α≥1. For α=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for α=2 it coincides with the typical Poisson-Voronoi cell. Let pn(α) be the probability for the zero-cell to have n sides. We construct the asymptotic expansion of log pn(α) up to terms that vanish as n→∞. Our methods are nonrigorous but of the kind commonly accepted in theoretical physics as leading to exact results. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when α>1, but gets delocalized for the Crofton cell, α=1, which is a singular point of the parameter range. The large-n expansion of log pn(1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the typicaln-sided cell of a Poisson line tessellation.

On the spectral function of the Poisson-Voronoi cells

Abstract Denote by ϕ(t)=∑ n⩾1 e −λ n t , t>0 , the spectral function related to the Dirichlet Laplacian for the typical cell C of a standard Poisson–Voronoi tessellation in R d , d⩾2 . We show that the expectation E ϕ(t) , t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E ϕ(t) , when t→0+,+∞. In particular, we prove that the law of the first eigenvalue λ1 of C satisfies the asymptotic relation ln P {λ 1 ⩽t}∼−2 d ω d j (d−2)/2 d ·t −d/2 when t→0+, where ωd and j(d−2)/2 are respectively the Lebesgue measure of the unit ball in R d and the first zero of the Bessel function J(d−2)/2.

Asymptotic Methods for Random Tessellations

In this chapter, we are interested in two classical examples of random tessellations which are the Poisson hyperplane tessellation and Poisson–Voronoi tessellation. The first section introduces the main definitions, the application of an ergodic theorem and the construction of the so-called typical cell as the natural object for a statistical study of the tessellation. We investigate a few asymptotic properties of the typical cell by estimating the distribution tails of some of its geometric characteristics (inradius, volume, fundamental frequency). In the second section, we focus on the particular situation where the inradius of the typical cell is large. We start with precise distributional properties of the circumscribed radius that we use afterwards to provide quantitative information about the closeness of the cell to a ball. We conclude with limit theorems for the number of hyperfaces when the inradius goes to infinity.

Refined convergence for the Boolean model

In Michel and Paroux (2003) the authors proposed a new proof of a well-known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process (see, e.g. Hall (1985)). In this paper we investigate the second-order term in this convergence when the two-dimensional Boolean model and the Poisson line process are coupled on the same probability space. We consider the particular case where the grains are discs with random radii. A precise coupling between the Boolean model and the Poisson line process is first established. A result of directional convergence in distribution for the difference of the two sets involved is then derived. Eventually, we show the convergence of the process, measuring the difference between the two random sets, once rescaled, as a function of the direction.