Fractal properties of Aldous-Kendall random metric

Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric $T$ on ${\mathbb{R}}^d$, for which the distance between points is given by the optimal connection time, when travelling on the road network generated by a Poisson process of lines with a speed limit. In this paper, we look into some fractal properties of that random metric. In particular, although almost surely the metric space $({\mathbb{R}}^d,T)$ is homeomorphic to the usual Euclidean ${\mathbb{R}}^d$, we prove that its Hausdorff dimension is given by $(\gamma -1)d\mathrm{/}(\gamma -d)>d$, where $\gamma >d$ is a parameter of the model; which confirms a conjecture of Kahn. We also find that the metric space $({\mathbb{R}}^d,T)$ equipped with the Lebesgue measure exhibits a multifractal property, as some points have untypically big balls around them.