graph.

The fact that large deviation regimes may be dominated by configurations with a small number of very large contributions was known relatively early in the history of large deviation theory: a survey by Nagaev [@Nagaev], summarizing a series of papers written in the Soviet Union in the 1960’s and 70’s, includes this observation for sums of i.i.d. random variables with stretched-exponential tails as a corollary. In our context, the natural combinatorial structure for creating many edges with a small number of edges is a “giant clique." The clique number, the (typical) size of the largest clique of the random geometric graph, falls under the general class scan statistics, has been shown to focus on two values with high probability for certain values of the threshold parameter $r$ (see [@Penrose4], [@Muller]); however, these works do not explore the large deviation regime. Our work uses techniques from large deviations, concentration inequalities, convex analysis, and geometric measure theory. A key component in the proof is a technique for proving localization that has previously appeared in [@Talagrand1] and [@Chatterjee].

Main result
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Let $\chi_n$ be a Poisson Point Process of intensity $n$ on the $d$-dimensional unit torus $\mathbb{T}^d = [0,1]^d$. Let $N := 