多面体随机样本中的最大最近邻链路和连通性阈值
Largest nearest-neighbour link and connectivity threshold in a polytopal random sample
摘要：
设$X_1,X_2, \ldots $是凸多面体区域$A \subset \mathbb{R}^d$中独立同分布的随机点.将最大最近邻链路$L_n$定义为最小$r$，使得$\mathcal X_n:=\{X_1,\ldots,X_n\}$的每个点在距离$r$内具有另一个这样的点。在大$n$极限下得到了$L_n$的一个强大数定律.相关阈值，连通性阈值$M_n$，是使得随机几何图$G(\mathcal X_n, r)$连通的最小$r$。我们表明，作为$n \to \infty$，几乎可以肯定的是，$nL_n^d/\log n$往往是一个限制，这取决于几何形状的$A$，和$nM_n^d/\log n$往往是相同的限制。
英文摘要：
Let $X_1,X_2, \ldots $ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.
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