Spectra of high-dimensional sparse random geometric graphs

Abstract

We determine the limiting empirical spectral distribution of sparse high-dimensional random geometric graphs. The vertices are independent uniform points on the unit sphere $S^{d-1}$, and two vertices are joined when their inner product exceeds a threshold chosen to give edge density $p$. The edges therefore have the same marginal probabilities as in an Erd\H{o}s--R\'enyi graph, but the latent geometry introduces dependence among them. We show that these correlations are asymptotically invisible to the global spectrum in two sparse regimes. If $p\to0$, $np\to\infty$, and $d=\Omega(np\log(1/p))$, then the empirical spectral distribution of $A/\sqrt{np}$ converges in probability to the semicircle law. If $p=\alpha/n$ for a fixed $\alpha>0$ and $d=\omega(\log n)$, then the empirical spectral distribution of $A/\sqrt{\alpha}$ converges in probability to the limiting spectral distribution of $\mathcal G(n,\alpha/n)$. The proof combines the moment method with a cluster expansion that decomposes geometric dependence into weak local interactions, allowing us to control every fixed walk pattern in the moment calculation.