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 $\langle v_i, v_j \rangle \geq \tau$ so that $p=\mathbb P(\langle v_i,v_j\rangle \geq \tau)$. This model defines a nonlinear random matrix ensemble with dependent entries. We show that if $d =\omega( np\log^{2}(1/p))$ 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 $p=\alpha/n$, we further show that if $d=\omega(\log^2(n))$ the limiting spectral distribution of $A$ in $G(n,d, \alpha/n)$ coincides with that of the Erd\H{o}s-R\'{e}nyi graph $ G(n,\alpha/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 $\mathbb E \mathrm{tr}(A^k)$. A refined high trace analysis further yields a near-optimal bound on the second eigenvalue when $np=\Omega(\log^4 (n))$, removing technical conditions previously imposed in (Liu et al. 2023).