fixed constant $\delta^* >0$ such that, for all sufficiently large $n$, $$\label{rcond}
n^{ (\delta^*-2)/d } \leq r_n \leq n^{- \delta^*/d}$$ The lower bound ensures that the expected number of edges is polynomially large in $n$; the upper bound excludes the possibility of $r_n$ vanishing as the inverse of a polynomial in $\log n$ (for example), but is not much stronger than the initial requirement of $r_n$ vanishing in as $n$ grows. We define a parameter $p$ as $$\label{pdef}
p : = \displaystyle \lim_{n \rightarrow \infty} \frac{ \log \mu_n}{\log n}\, ,$$ implicitly assuming that the limit exists; we will continue to assume this for the rest of the paper.

![An instance of the random geometric graph $G(\chi_{150},0.1)$, with respect to the Euclidean norm. The graph has 148 vertices and 343 edges. The gray area is the white unit square translated, to show periodicity](rgg2.png)

This ensures that $\mu_n = f(n) n^p$, where $f$ varies more slowly than any polynomial or rational function in $n$. Notice that $p$ depends only on the tail behavior of $r$, and is completely independent of $n$. Furthermore, $$\label{pbounds}
\delta^* \leq p \leq 2 - \delta^*\, ,$$ thanks to .

The following theorem is the main result of 