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

Abstract

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 $np=\Omega (\log n)$, the spherical adjacency matrix satisfies, with high probability,$\Vert A-\mathbb{E}A{\Vert }_{\mathrm{op}}=O\left(\sqrt{np\log n}+np\tau \right)$, 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 $np\gg \log n$, vector and relative Gram-matrix errors vanish for$\log (1/p)\ll d\ll np\log (1/p)/\log n$ in the spherical model and ${\log }^2(1/p)\log n\ll d\ll np\log (1/p)/\log n$ 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.