satisfies an upper tail large deviation principle with speed $\sqrt{\mu_n} \log n$ and rate function $I(x)$.

Note that this rate function, like all other claims in this paper, is only valid for events in which the number of edges exceeds its mean. The lower tail of the edge count is likely to satisfy Poisson-like statistics, and hence the speed of its large deviation probability is expected to be of order $\mu_n$, not $\sqrt{\mu_n} \log n$.

Before we go on, we’d like to comment on the heuristic formulation we presented earlier. First, since the rate function of \[thmLDP\] is strictly increasing, we know that, conditional on $\{|E| > (1 + \delta) \mu_n\}$, the event $$\left\{(1 + \delta') \mu_n > |E| > (1 + \delta) \mu_n\right\}$$ occurs with high probability for any $\delta' > \delta$ and $n$ sufficiently large. If we take $S = A$ in the first stipulation of \[thm1\], we see that the ball $A$ of diameter $r_n$ includes at least $\delta \mu_n(1 - \varepsilon)$ edges of the random geometric graph. Since $\varepsilon$ and $\delta' -
\delta$ are arbitrarily close to zero, we recover our original claim that ‘almost all extra edges are between points in $A$’. In terms of 