Spectra of high-dimensional sparse random geometric graphs

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 $⟨v_i,v_j⟩\ge \tau$ so that $p=\mathbb{P}(⟨v_i,v_j⟩\ge \tau )$. This model defines a nonlinear random matrix ensemble with dependent entries. We show that if $d=\omega (np{\log }^2(1\mathrm{/}p))$ and $np\to \mathrm{\infty }$, 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 $p=\alpha \mathrm{/}n$, we further show that if $d=\omega ({\log }^2(n))$ the limiting spectral distribution of $A$ in $G(n,\alpha \mathrm{/}n)$ coincides with that of the Erdős-Rényi graph $G(n,\alpha \mathrm{/}n)$. 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 $np=\mathrm{\Omega }({\log }^4(n))$, 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.