k-近邻随机几何图模型中的共享性
Sharpness in the k-nearest neighbours random geometric graph model
摘要：
令$S_{n,k}$表示通过根据强度为1的泊松过程将点放置在面积为$n$的正方形框中并将每个点连接到其$k$最近邻居而获得的随机几何图。Balister、Bollobás、Sarkar和Walters指出，对于每个$0< ε<1$和所有足够大的$n$，都存在$C=C(ε)$，使得只要$S_{n,k}$连通的概率至少为$ε$，那么$S_{n,k+C}$连通的概率至少为$1-ε$。本文证明了这一猜想。作为推论，我们证明了存在一个常数$C'$，使得当$k=k(n)$是一个整数序列时，当$n$趋于无穷大时，$S_{n,k(n)}$连通的概率趋于1，那么对于任何具有$s(n)=o(\log n)$的$s(n)$，$S_{n,k(n)+C's\log \log n}$是$s$连通的概率趋于1.这证明了Balister，Bollobás，Sarkar和Walters的另一个猜想.
英文摘要：
Let $S_{n,k}$ denote the random geometric graph obtained by placing points in a square box of area $n$ according to a Poisson process of intensity 1 and joining each point to its $k$ nearest neighbours. Balister, Bollobás, Sarkar and Walters conjectured that for every $0< ε<1$ and all $n$ sufficiently large there exists $C=C(ε)$ such that whenever the probability $S_{n,k}$ is connected is at least $ε$ then the probability $S_{n,k+C}$ is connected is at least $1-ε$. In this paper we prove this conjecture. As a corollary we prove that there is a constant $C'$ such that whenever $k=k(n)$ is a sequence of integers such that the probability $S_{n,k(n)}$ is connected tends to one as $n$ tends to infinity, then for any $s(n)$ with $s(n)=o(\log n)$, the probability that $S_{n,k(n)+C's\log \log n}$ is $s$-connected tends to one This proves another conjecture of Balister, Bollobás, Sarkar and Walters.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
