On the components of random geometric graphs in the dense limit

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\mathbf{R}}^d,d\ge 2$, with $C^2$ boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of components of this graph, and $R_n$ the number of vertices not in the giant component. Let $S_n$ be the number of isolated vertices. We show that if $r_n$ is chosen so that $n{r}_n^d$ tends to infinity but slowly enough that $\mathbf{E}[S_n]$ also tends to infinity, then $K_n$, $R_n$ and $S_n$ are all asymptotic to ${\mu }_n$ in probability as $n\to \mathrm{\infty }$ where (with $\mathrm{\mid }A\mathrm{\mid }$, ${\theta }_d$ and $\mathrm{\mid }\mathrm{\partial }A\mathrm{\mid }$ denoting the volume of $A$, of the unit $d$-ball, and the perimeter of $A$ respectively) ${\mu }_n:=n{e}^{-\pi n(r_n{)}^d\mathrm{/}\mathrm{\mid }A\mathrm{\mid }}$ if $d=2$ and ${\mu }_n:=n{e}^{-{\theta }_{d}n(r_n{)}^d\mathrm{/}\mathrm{\mid }A\mathrm{\mid }}+{\theta }_{d-1}^{-1}\mathrm{\mid }\mathrm{\partial }A\mathrm{\mid }(r_n{)}^{1-d}{e}^{-{\theta }_{d}n(r_n{)}^d\mathrm{/}(2\mathrm{\mid }A\mathrm{\mid })}$ if $d\ge 3$. We also give variance asymptotics and central limit theorems for $K_n$ and $R_n$ in this limiting regime when $d\ge 3$, and for Poisson input with $d\ge 2$. We extend these results (substituting $\mathbf{E}[S_n]$ for ${\mu }_n$) to a class of non-uniform distributions on $A$.