(such as generalized Cantor sets) or have boundaries that take up a large amount of space. We also force $S$ to be large to make sure it cannot pick up the lower order uneveness in the conditional process. Probing the Poisson Point Process with such sets is sufficient to show that graph will appear uniform, up to sets of measure equal to arbitrarily small multiples of $\tau_n$.

As a consequence of Theorem \[thm1\], we prove that the upper tail of the edge count of random geometric graphs satisfies a large deviation principle. Recall that $(|E|-\mu_n)/\mu_n$ satisfies an [*upper tail*]{} large deviation principle with speed $f(n)$ and rate function $I(x)$ if, for any closed set $F \subset (0,\infty)$, $$\displaystyle \limsup_{n\to \infty}\frac{1}{f(n)} \log \mathbb{P}\left(\frac{|E| - \mu_n}{\mu_n} \in F \right) \leq -\displaystyle \inf_{x \in F} I(x)\,
,$$ and for any open set $G \subset [0,\infty)$, $$\displaystyle \liminf_{n\to\infty} \frac{1}{f(n)} \log \mathbb{P}\left(\frac{|E| - \mu_n}{\mu_n} \in G \right) \geq - \displaystyle \inf_{x \in G} I(x)\,
.$$

\[thmLDP\] Let $G(\chi_n,r_n)$ be a random geometric graph model on the d-dimensional torus, with the same assumptions as in Theorem \[thm1\]. Define $$I(x) := \left(\frac{2-p}{2}\right) \sqrt{2 x}\, ,$$ where $p$ is defined as in . Then the random variable $(|E|-\mu_n)/\mu_n$ 