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 Sn,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 Sn,k is asymptotic to its mean, and prove Poisson and normal approximation results for Sn,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.