Coverage of the unit cube by dynamic Boolean models

Abstract

Motivated by peer-to-peer telecommunication, we study a dynamic Boolean model. We define a Poisson number of random lines through the $(d-1)$-dimensional base of a $d$-dimensional unit cube and dilate them to define cylinders. Letting $\rho$ be the expected number of cylinders, the random variable of interest is the coverage radius $R_\rho$, which is the cylinder radius required to cover the $d$-dimensional unit cube. We show that $R_\rho^{d-1}$ is of the order $\log \rho / \rho$ with high probability as $\rho$ tends to infinity. We also consider alternative dynamics resulting in generalized cylinders that are generated by dilating the trajectories of stochastic processes, in particular Brownian motions. This leads to a coverage radius of the same order.