Euclidean and chemical distances in ellipses percolation

The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $\alpha$ controls the tail distribution of the major axis’ distribution and we focus on the regime $\alpha \in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between points $x$ and $y$ behaves roughly as $c\log \log \mathrm{\mid }x-y\mathrm{\mid }$.