{"ID":2896420,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.06556","arxiv_id":"2507.06556","title":"Spectra of high-dimensional sparse random geometric graphs","abstract":"We analyze the spectral properties of the high-dimensional random geometric graph $G(n, d, p)$, formed by sampling $n$ i.i.d vectors $\\{v_i\\}_{i=1}^{n}$ uniformly on a $d$-dimensional unit sphere and connecting each pair $\\{i,j\\}$ whenever $\\langle v_i, v_j \\rangle \\geq τ$ so that $p=\\mathbb P(\\langle v_i,v_j\\rangle \\geq τ)$. This model defines a nonlinear random matrix ensemble with dependent entries. We show that if $d =ω( np\\log^{2}(1/p))$ and $np\\to\\infty$, the limiting spectral distribution of the normalized adjacency matrix $\\frac{A}{\\sqrt{np(1-p)}}$ is the semicircle law. To our knowledge, this is the first such result for $G(n, d, p)$ in the sparse regime. In the constant sparsity case $p=α/n$, we further show that if $d=ω(\\log^2(n))$ the limiting spectral distribution of $A$ in $G(n,α/n)$ coincides with that of the Erdős-Rényi graph $G(n,α/n)$. Our approach combines the classical moment method in random matrix theory with a novel recursive decomposition of closed-walk graphs, leveraging block-cut trees and ear decompositions, to control the moments of the empirical spectral distribution. A refined high trace analysis further yields a near-optimal bound on the second eigenvalue when $np=Ω(\\log^4 (n))$, removing technical conditions previously imposed in (Liu et al. 2023). As an application, we demonstrate that this improved eigenvalue bound sharpens the parameter requirements on $d$ and $p$ for spontaneous synchronization on random geometric graphs in (Abdalla et al. 2024) under the homogeneous Kuramoto model.","short_abstract":"We analyze the spectral properties of the high-dimensional random geometric graph $G(n, d, p)$, formed by sampling $n$ i.i.d vectors $\\{v_i\\}_{i=1}^{n}$ uniformly on a $d$-dimensional unit sphere and connecting each pair $\\{i,j\\}$ whenever $\\langle v_i, v_j \\rangle \\geq τ$ so that $p=\\mathbb P(\\langle v_i,v_j\\rangle \\g...","url_abs":"https://arxiv.org/abs/2507.06556","url_pdf":"https://arxiv.org/pdf/2507.06556v4","authors":"[\"Yifan Cao\",\"Yizhe Zhu\"]","published":"2025-07-09T05:23:13Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.CO\",\"math.ST\"]","methods":"[]","has_code":false}