Related papers: Coverage of the unit cube by dynamic Boolean model…

In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set $A \subset \mathbb{R}^d.$ This Poisson process of cylinders is…

We consider two dynamical variants of Dvoretzky's classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent…

We derive fundamental asymptotic results for the expected covering radius $\rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For…

We consider a dynamic network in continuum time and space in which nodes, with initial locations given by a Poisson point process, move according to i.i.d. isotropic $\alpha$-stable processes. Each node is additionally equipped with an…

We study visibility inside the vacant set of three models in $\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Boolean model. For each of them, we obtain sharp asymptotic bounds on the…

We consider the {following} coverage model on $\mathbb{N}$. For each site $i\in \mathbb{N}$ we associate a pair $(\xi_i, R_i)$ where $\{\xi_0, \xi_1, \ldots \}$ is a 1-dimensional {undelayed} discrete renewal point process and…

In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there…

Two models of random cones in high dimensions are considered, together with their duals. The Donoho-Tanner random cone $D_{n,d}$ can be defined as the positive hull of $n$ independent $d$-dimensional Gaussian random vectors. The Cover-Efron…

We compute the time-dependent coverage in the random sequential adsorption of aligned d-dimensional cubes in $R^d$ using time-series expansions. The seventh-order series in 2, 3 and 4 dimensions is resummed in order to predict the coverage…

Consider a Boolean model $\Sigma$ in $\R^d$. The centers are given by a homogeneous Poisson point process with intensity $\lambda$ and the radii of distinct balls are i.i.d.\ with common distribution $\nu$. The critical covered volume is…

Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb{R}^d$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a…

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite…

Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d \geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r \geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with finite…

We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We…

This paper develops a framework to study multi-hop relaying in a vehicular network consisting of vehicles and Road Side Units (RSUs), and the effect of this relaying on the network coverage and the communication delay. We use a stochastic…

The union of the particles of a stationary Poisson process of compact (convex) sets in Euclidean space is called Boolean model and is a classical topic of stochastic geometry. In this paper, Boolean models in hyperbolic space are…

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood…

We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of…

We consider a Poisson point process on the space of lines in R^d, where a multiplicative factor u>0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius…

We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length $L$, density $\rho$, dimension $d$ and jump density $\varphi$, one samples $\rho L^d$ particles in a…