On the edge eigenvalues of sparse random geometric graphs

Abstract: In this paper, we study the edge eigenvalues of random geometric graphs (RGGs) generated by multivariate Gaussian samples in the sparse regime under a broad class of distance metrics. Previous work on edge eigenvalues under related setups has relied on methods based on integral operators or the Courant-Fischer min-max principle with interpolation. However, these approaches typically require either a dense regime or sampling distributions that are compactly supported and non-vanishing, and therefore cannot be generalized to our setting. We introduce a two-step smoothing-matching argument. First, we construct a smoothed empirical operator from the given RGG. We then match its edge eigenvalues to those of a continuum limit operator via a counting argument. We show that, after proper normalization, the first few nontrivial edge eigenvalues of the RGG converge with high probability to those of a differential operator, which can be computed explicitly through a simple second-order linear partial differential equation. To the best of our knowledge, these are the first results on the edge eigenvalues of RGGs generated from samples with unbounded support and vanishing density functions.