论文标题
Aldous-Kendall随机度量的分形特性
Fractal properties of Aldous-Kendall random metric
论文作者
论文摘要
肯德尔（Kendall）调查了Aldous建议的比例不变的随机空间网络模型，在$ \ Mathbb {r}^d $上构建了一个随机度量$ t $，为此，当点在路线网络上，由具有速度限制限制的线路的路线过程产生的路网络时，点之间的距离是由最佳连接时间给出的。在本文中，我们研究了该随机度量的某些分形特性。特别是，尽管几乎可以肯定的是，$ \ left（\ mathbb {r}^d，t \ right）$对通常的Euclidean $ \ mathbb {r}^d $是同型，但我们证明其hausdorff dimension的hausdorff dimension $（γ-1）d/γ-d/γ-d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d of y $γ;这证实了卡恩的猜想。我们还发现，配备了Lebesgue Measure的度量空间$ \ left（\ Mathbb {r}^d，t \ right）$具有多重属性的特性，因为某些点在它们周围有毫无疑问的大球。
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.
