Spectral Concentration and Recovery in Sparse High-Dimensional Random Geometric Graphs

We study sparse threshold random geometric graphs generated by high-dimensional spherical or Gaussian latent vectors. Although each edge has marginal probability p, shared latent variables make the adjacency entries dependent. At the connectivity scale np= (n), the spherical adjacency matrix satisfies, with high probability, A- E A op=O (sqrt np n +np tau), where tau is the cap threshold; an analogous estimate holds for Gaussian vectors after controlling radial fluctuations. This sharpens the spectral bound in Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions and strengthens the global-synchronization guarantee of Abdalla, Bandeira, and Invernizzi (2024) for the homogeneous Kuramoto model. The leading eigenspace also estimates the latent geometry. When np n, vector and relative Gram-matrix errors vanish for (1/p) d np (1/p)/ n in the spherical model and 2(1/p) n d np (1/p)/ n in the Gaussian model, improving the recovery conditions of Li and Schramm (2023). For the Gaussian mixture block model introduced there, a polynomial-time semidefinite program gives, to our knowledge, the first exact-recovery guarantee at the connectivity scale in a moderate-separation regime. At much larger separation, fixed edge density creates isolated vertices and makes exact recovery impossible. Our reusable decoupling and matrix concentration framework avoids trace-moment methods and applies broadly to random graph models with latent vectors.