2022/07/07 by Blanc, Guillaume
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2207.03349
Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric T on ℝd, 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 (ℝd,T) is homeomorphic to the usual Euclidean ℝd, we prove that its Hausdorff dimension is given by (γ-1)d/(γ-d)>d, where γ>d is a parameter of the model; which confirms a conjecture of Kahn. We also find that the metric space (ℝd,T) equipped with the Lebesgue measure exhibits a multifractal property, as some points have untypically big balls around them.