On k-clusters of high-intensity random geometric graphs

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 d -dimensional Euclidean space, 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.