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 $\langle v_i, v_j \rangle \geq τ$ 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.

💡 Research Summary

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This paper investigates the spectral properties of high‑dimensional random geometric graphs (RGGs) denoted by (G(n,d,p)). The model is defined by sampling (n) independent vectors uniformly from the unit sphere (\mathbb{S}^{d-1}) and connecting vertices (i) and (j) whenever their inner product exceeds a threshold (\tau) chosen so that the edge probability equals a prescribed value (p). Consequently the expected degree is (np), and the graph becomes sparse when (p) tends to zero.

The authors address two major challenges that simultaneously appear in this setting: (i) the adjacency matrix is a non‑linear random matrix because the entry (A_{ij}=1{\langle v_i,v_j\rangle\ge\tau}) is a non‑linear function of the underlying Gaussian‑like vectors, and (ii) the matrix is sparse, i.e., many entries are zero. Existing results on non‑linear random matrices typically assume a fixed, non‑vanishing function and dense regimes, while sparse random matrix theory usually relies on independence of entries. The combination of non‑linearity and sparsity makes the analysis considerably more delicate.

Main Contributions

1. Semicircle Law for Dense‑to‑Sparse Regime (Theorem 2.1).
   Under the conditions
   \