  In this article, we analyze the limiting eigenvalue distribution (LED) of
random geometric graphs (RGGs). The RGG is constructed by uniformly
distributing $n$ nodes on the $d$-dimensional torus $\mathbb{T}^d \equiv [0,
1]^d$ and connecting two nodes if their $\ell_{p}$-distance, $p \in [1,
\infty]$ is at most $r_{n}$. In particular, we study the LED of the adjacency
matrix of RGGs in the connectivity regime, in which the average vertex degree
scales as $\log\left( n\right)$ or faster, i.e., $\Omega \left(\log(n)
\right)$. In the connectivity regime and under some conditions on the radius
$r_{n}$, we show that the LED of the adjacency matrix of RGGs converges to the
LED of the adjacency matrix of a deterministic geometric graph (DGG) with nodes
in a grid as $n$ goes to infinity. Then, for $n$ finite, we use the structure
of the DGG to approximate the eigenvalues of the adjacency matrix of the RGG
and provide an upper bound for the approximation error.
