Aldous-Kendall随机度量的分形性质
Fractal properties of Aldous-Kendall random metric
摘要：
研究了Aldous提出的尺度不变随机空间网络模型，Kendall在由限速线路组成的泊松过程生成的道路网络上，构造了以最优连接时间给出点间距离的随机度量$\mathbb{R}^d$上的$T$.本文研究了这种随机度量的一些分形性质。特别是，虽然几乎肯定的度量空间$\left(\mathbb{R}^d,T\right)$同胚通常的欧几里德$\mathbb{R}^d$，我们证明了它的豪斯多夫维数是由$(γ-1)d/(γ-d)>d$，其中$γ>d$是一个参数的模型，这证实了卡恩的猜想。我们还发现，度量空间$\left(\mathbb{R}^d,T\right)$配备了勒贝格测度表现出多重分形性质，因为一些点有不寻常的大球在他们周围。
英文摘要：
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 $\left(\mathbb{R}^d,T\right)$ is homeomorphic to the usual Euclidean $\mathbb{R}^d$, 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 $\left(\mathbb{R}^d,T\right)$ equipped with the Lebesgue measure exhibits a multifractal property, as some points have untypically big balls around them.
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