论文标题
在$ k $ - 高强度随机几何图的clusters上
On $k$-clusters of high-intensity random geometric graphs
论文作者
论文摘要
让$ k，d $为正整数。我们确定了一系列常数序列，这些常数渐近地是在$ d $二维的泊松布尔型中的群集群集在固定半径的球中，因为强度很大，固定半径为$ k $。 Using this, we determine the asymptotics of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a random geometric graph on $n$ uniformly distributed vertices in a smoothly bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that the expected degree grows slowly as $n$ becomes large (the so-called mildly dense limiting regime).我们还表明，$ s_ {n，k} $的差异对其均值渐近，并且在此限制方案中证明了$ s_ {n，k} $的Poisson和正常近似结果。我们为相应的泊松过程提供了类似的结果（即带有泊松数的点）。 我们还在所谓的轻度稀疏限制制度中给出了类似的结果，其中选择了$ r（n）$，因此随着$ n $变大，预期的学位衰减缓慢至零。
Let $k,d $ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$-dimensional Poisson Boolean model with balls of fixed radius is of order $k$, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a random geometric graph on $n$ uniformly distributed vertices in a smoothly bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that the expected degree grows slowly as $n$ becomes large (the so-called mildly dense limiting regime). We also show that the variance of $S_{n,k}$ is asymptotic to its mean, and prove Poisson and normal approximation results for $S_{n,k}$ in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points). We also give similar results in the so-called mildly sparse limiting regime where $r(n)$ is chosen so the expected degree decays slowly to zero as $n $ becomes large.
