d维最近邻树的树结构
Siblings in d-dimensional nearest neighbour trees
摘要：
在$d$维球体上拾取一系列均匀点。然后，将第$n$个点链接到过去到达的最近点。这在$d$维球面上构造了一个被称为最近邻树的标记树。这些树与随机递归树共享一些属性：最后到达节点的高度，根的平均度等。相反，叶子的数量似乎取决于维数$d$，但还没有这样的属性被证明。在这篇文章中，我们证明了兄弟姐妹的平均数取决于$d$。特别是，我们给出了这个数字的显式计算。在维数$1$中，它是$1 + \ln 2$，在任何维数$d$中，它都有一个显式积分形式，但不幸的是，它没有给出一个显式数字。然而，我们发现，当$d \to \infty$以$\sqrt{3}/2$的速率指数快速时，它收敛到$2$。为了证明这些结果，我们研究了这些树的局部极限，并对$d$维中的两个球的相交做了一些精细的计算。特别是，我们得到了一个非平凡的上界，这些交叉在某些精确的情况下。
英文摘要：
Pick a sequence of uniform points on the $d$-dimensional sphere. Then, link the $n$th point to its closest one that arrives in the past. This constructs a labelled tree called the nearest neighbour tree on the $d$-dimensional sphere. These trees share some properties with the random recursive tree: the height of the last arrival node, the mean degree of the root, etc. On the contrary, the number of leaves seems to depend on dimension $d$, but no such properties have been proved yet. In this article, we prove that the mean number of siblings depends on $d$. In particular, we give explicit calculations of this number. In dimension $1$, it is $1 + \ln 2$ and, in any dimension $d$, it has an explicit integral form, but unfortunately, it does not give an explicit number. Nevertheless, we show that it converges to $2$ when $d \to \infty$ exponentially quick at a rate of $\sqrt{3}/2$. To prove these results, we look at the local limit of those trees and we do some fine computations about the intersection of two balls in dimension $d$. In particular, we obtain a non-trivial upper bound for those intersections in some precise cases.
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