The geometry of the giant component of random geometric graphs
Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\Delta$ denote the expected average degree of the graph. As is the case for Erd\H{o}s-R\'enyi graphs, there is a critical value $\Delta_c$, depending only on the dimension of $M$, such that if $\Delta > \Delta_c$ then $G_M(n;r)$ has a giant component. We show that whenever $\Delta > \Delta_c$, the giant component of $G_M(n;r)$, equipped with the graph distance, converges to the underlying manifold $M$ in the Gromov-Hausdorff distance after rescaling by an appropriate deterministic factor. Our result holds for $\Delta$ depending on $n$ as well, provided $\Delta = o(n)$ and $\Delta \geq \Delta_c + \varepsilon$ for any fixed $\varepsilon > 0$. As a consequence, we show that for any pair of non-isometric compact Riemannian manifolds $M_1$ and $M_2$, there is a polynomial-time algorithm that distinguishes random geometric graphs on $M_1$ and $M_2$ throughout this regime of $\Delta.$ In the thermodynamic regime -- i.e.\ when $\Delta$ is constant -- our results appear to be new even in the classical cases where $M$ is a sphere or a torus. Our proof makes use of techniques from first-passage percolation which allow us to understand the long-range behavior of the graph distance on small, approximately Euclidean patches of $M$, together with global arguments that glue these local estimates into a global description.