Robustness in the Poisson Boolean model with convex grains

Abstract

We study the Poisson Boolean model where the grains are random convex bodies with a rotation-invariant distribution. A grain distribution is dense if the union of the grains covers the entire space and robust if the union of the grains has an unbounded connected component irrespective of the intensity of the underlying Poisson process. If the grains are balls of random radius, then density and robustness are equivalent, but in general this is not the case. We show that in any dimension $d\ge 2$ there are grain distributions that are robust but not dense and give criteria for density, robustness and non-robustness of a grain distribution. For robustness, we provide a criterion that requires only information on the tail index of the marginal diameter distributions of the grains, as well as criteria that apply when diameters are sufficiently positively correlated. We also show the existence of grains, where the marginal distribution of the diameters does not provide sufficient information to determine whether the grain distribution is robust.

Major revision of previous version, including correction of statements of the theorems