|\chi_n|$. Recall that $N$ is a Poisson random variable with mean $n$, and conditional on $N$, $\chi_n$ is just a set of $N$ points chosen independently and uniformly at random. Let $r_n$ be a positive sequence that decreases to $0$ as $n\to \infty$, and $\| \cdot \|$ be some norm on $\mathbb{R}^d$ that induces a translation-invariant metric on $\mathbb{T}^d$. We define the random geometric graph $G(\chi_n,r_n) : = (V,E)$, where $V = \chi_n = \{v_1,\ldots, v_N\}$, enumerated arbitrarily, and $E$ is the set of unordered pairs $\{i,j\}$ such that $\| v_i - v_j\| \leq r_n$. Figure 1 shows a particular instance of $G(\chi_{150},0.1)$.

Letting $1_{i,j}$ be the indicator that there is an edge between $v_i$ and $v_j$, we can calculate the expected value of $|E|$, the number of edges in the graph: $$\begin{aligned}
\mathbb{E}(|E|) &= \mathbb{E}\left(\displaystyle \sum_{1\le i<j\le N} 1_{i,j} \right) = \mathbb{E}\left[ {N \choose 2} \mathbb{E}(1_{1,2} \mid N) \right]\\
&= \frac{n^2}{2} \mathbb{P}(\|v_1-v_2\|\le r_n).\end{aligned}$$ By the symmetry of the torus, $\mathbb{P}(\|v_1-v_2\|\le r_n)$ is simply $\nu r_n^d$, where $\nu$ is the volume of the unit ball of the given norm, $r_n$. Thus, $$\mathbb{E}(|E|) = \frac{n^2 \nu r_n^d}{2} =: \mu_n.$$ For the rest of the article, we suppose the existence of a 