Marilyn Korfhage | Searcharxiv

arXiv2026

Robustness in the Poisson Boolean model with convex grains

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 $d\ge2$ 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.

math.PR↗

arXiv2025

Chemical distance in the Poisson Boolean model with regularly varying diameters

We study the Poisson Boolean model with convex bodies which are rotation-invariant distributed. We assume that the convex bodies have regularly varying diameters with indices $-α_1\geq \dots\geq-α_d$ where $α_k >0$ for all $k\in\{1,\dots,d\}.$ It is known that a sufficient condition for the robustness of the model, i.e. the union of the convex bodies has an unbounded connected component no matter what the intensity of the underlying Poisson process is, is that there exists some $k\in\{1,\dots,d\}$ such that $α_k<\min\{2k,d\}$. To avoid that this connected component covers all of $\mathbb{R}^d$ almost surely we also require $α_k> k$ for all $k\in\{1,\dots,d\}$. We show that under these assumptions, the chemical distance of two far apart vertices $\mathbf{x}$ and $\mathbf{y}$ behaves like $c\log\log|x-y|$ as $|x-y|\rightarrow \infty$, with an explicit and very surprising constant $c$ that depends only on the model parameters. We furthermore show that if there exists $k$ such that $α_k\leq k$, the chemical distance is smaller than $c\log\log|x-y|$ for all $c>0$ and that if $α_k\geq\min\{2k,d\}$ for all $k$, it is bigger than $c\log\log|x-y|$ for all $c>0$.

math.PR↗