Variance asymptotics and scaling limits for Gaussian Polytopes

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input

Variance Asymptotics and Scaling Limits for Random Polytopes

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$ for the boundary of K $\lambda$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $\lambda$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $\lambda$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity \sqrt de dh dhdv

Some Classical Problems in Random Geometry

International audienceThis chapter is intended as a first introduction to selected topics in random geometry. It aims at showing how classical questions from recreational mathematics can lead to the modern theory of a mathematical domain at the interface of probability and geometry. Indeed, in each of the four sections, the starting point is a historical practical problem from geometric probability. We show that the solution of the problem, if any, and the underlying discussion are the gateway to the very rich and active domain of integral and stochastic geometry, which we describe at a basic level. In particular, we explain how to connect Buffon’s needle problem to integral geometry, Bertrand’s paradox to random tessellations, Sylvester’s four-point problem to random polytopes and Jeffrey’s bicycle wheel problem to random coverings. The results and proofs selected here have been especially chosen for non-specialist readers. They do not require much prerequisite knowledge on stochastic geometry but nevertheless comprise many of the main results on these models

Statistical and renewal results for the random sequential adsorption model applied to a unidirectional multicracking problem

AbstractWe work out a stationary process on the real line to represent the positions of the multiple cracks which are observed in some composites materials submitted to a fixed unidirectional stress ɛ. Our model is the one-dimensional random sequential adsorption. We calculate the intensity of the process and the distribution of the inter-crack distance in the Palm sense. Moreover, the successive crack positions of the one-sided process (denoted by Xiɛ, i⩾1) are described. We prove that the sequence {(Xiɛ,Yiɛ),1⩽i⩽n} is a “conditional renewal process”, where Yiɛ is the value of the stress at which Xiɛ forms. The approaches “in the Palm sense” and “one-sided process” merge when n→+∞. The saturation case (ɛ=+∞) is also investigated

La loi du plus petit disque contenant la cellule typique de Poisson–Voronoi

National audienc

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \R^d$ be a smooth convex set and let \P_\la be a Poisson point process on $\R^d$ of intensity \la. The convex hull of \P_\la \cap K is a random convex polytope K_\la. As \la \to \infty, we show that the variance of the number of $k$-dimensional faces of K_\la, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$

Refined convergence for the Boolean model

In a previous work, two of 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. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. 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 derived as well.Comment: 33 page