Localization in random geometric graphs with too many edges

We consider a random geometric graph G(.n,rn), given by connecting two vertices of a Poisson point process .n of intensity n on the d-dimensional unit torus whenever their distance is smaller than the parameter rn. The model is conditioned on the rare event that the number of edges observed, |E|, is greater than (1+.)E(|E|), for some fixed .>0. This article proves that upon conditioning, with high probability there exists a ball of diameter rn which contains a clique of at least 2.E(|E|)........(1..) vertices, for any given .>0. Intuitively, this region contains all the .excess. edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be nonconvex.