Chemical distance in the Poisson Boolean model with regularly varying diameters

Abstract: 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 $-\alpha_1\geq \dots\geq-\alpha_d$ where $\alpha_k >0$ 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 $\alpha_k<\min{2k,d}$. To avoid that this connected component covers all of $\mathbb{R}^d$ almost surely we also require $\alpha_k> k$ 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 $\alpha_k\leq k$, the chemical distance is smaller than $c\log\log|x-y|$ for all $c>0$ and that if $\alpha_k\geq\min{2k,d}$ for all $k$, it is bigger than $c\log\log|x-y|$ for all $c>0$.