Covering the whole space with Poisson random balls

Overview & Synopsis

We consider Poisson random balls, with the pair (center, radius) being given by a Poisson point process. According to the intensity measure of the Poisson process, we investigate the eventuality of covering the whole space with the union of the balls. We exhibit a disjunction phenomenon between the coverage with large balls (low frequency) and the coverage with small balls (high frequency). Concerning the second type of coverage, we prove the existence of a critical regime which separates the case where coverage occurs a.s. and the case where coverage does not occur a.s. We give an explicit value of the critical intensity and we prove that the Hausdorff measure of the set of points which are not covered by the union of balls is linked with this value. We also compare with other critical regimes appearing in continuum percolation.