Robustness in the Poisson Boolean model with convex grains

TL;DR

This paper investigates the conditions under which a Poisson Boolean model with convex grains exhibits robustness and density, revealing that these properties can differ depending on grain distribution, especially beyond simple spherical cases.

Contribution

The authors establish criteria distinguishing robustness from density in the Poisson Boolean model with convex grains, including examples and sharpness of these criteria.

Findings

Robustness and density are equivalent for spherical grains.

Existence of grain distributions that are robust but not dense in higher dimensions.

Provided sharp criteria for density and robustness in convex grain models.

Abstract

We study the Poisson Boolean model where the grains are random convex bodies with a rotation-invariant distribution. We say that 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 $d\ge2$ there are grain distributions that are robust but not dense, and give general criteria for density, robustness and non-robustness of a grain distribution. We give examples which show that our criteria are sharp in many instances.