随机最近邻树的新结果
New results for the random nearest neighbor tree
AI总结：
该文研究$d$-NN随机树的结构性质，得到度序列的指数衰减与集中性、最大度对数阶等结果，还构造了其无穷局部极限并证明了局部有限、常返与连通性。
摘要：
本文研究维度为$d\in \mathbb N$的在线最近邻随机树（简称$d$-NN树），其定义如下：固定维度为$d$、面积为$n$的环面$\mathbb T^d_n$，赋予其由$\mathbb R^d$中欧氏度量导出的度量。随后，将$n$个顶点独立均匀随机地依次嵌入$\mathbb T^d_n$中，除第一个顶点外，每个顶点都连接到其（已嵌入的）最近邻，所得图记为$G_n$。\n 我们得到了多个关于$G_n$度序列的结果。首先，证明了$d$-NN树中度至少为$k\in \mathbb N$的顶点数通常随$k$指数衰减，且通过一个可能具有独立研究价值的新型Lipschitz型集中不等式实现了紧密集中。其次，得到$G_n$的最大度为对数阶。第三，给出了与维度无关的叶子数显式界，还给出了长度为2的路径数的估计。此外，证明了$G_n$中均匀选取顶点的高度通常为$(1+o(1))\log n$，$\mathbb T^d_n$的直径为$(2e+o(1))\log n$，且均与维度无关。\n 最后，定义了$G_n$的一个自然的无穷类似物$G_{\infty}$，并证明它对应有限图序列$(G_n)_{n \ge 1}$的局部极限。此外，几乎必然地证明了$G_{\infty}$是局部有限的，$G_{\infty}$上的简单随机游走是常返的，且$G_{\infty}$是连通的。
英文摘要：
In this paper, we study the online nearest neighbor random tree in dimension $d\in \mathbb N$ (called $d$-NN tree for short) defined as follows. We fix the torus $\mathbb T^d_n$ of dimension $d$ and area $n$ and equip it with the metric inherited from the Euclidean metric in $\mathbb R^d$. Then, embed consecutively $n$ vertices in $\mathbb T^d_n$ uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph $G_n$. We show multiple results concerning the degree sequence of $G_n$. First, we prove that typically the number of vertices of degree at least $k\in \mathbb N$ in the $d$-NN tree decreases exponentially with $k$ and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of $G_n$ is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in $G_n$ is $(1+o(1))\log n$ and the diameter of $\mathbb T^d_n$ is $(2e+o(1))\log n$, independently of the dimension. Finally, we define a natural infinite analog $G_{\infty}$ of $G_n$ and show that it corresponds to the local limit of the sequence of finite graphs $(G_n)_{n \ge 1}$. Moreover, we prove almost surely that $G_{\infty}$ is locally finite, that the simple random walk on $G_{\infty}$ is recurrent, and that $G_{\infty}$ is connected.
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