Spectra of high-dimensional sparse random geometric graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917261845987328 |
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| author | Cao, Yifan Zhu, Yizhe |
| author_facet | Cao, Yifan Zhu, Yizhe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectra of high-dimensional sparse random geometric graphs Cao, Yifan Zhu, Yizhe Probability Combinatorics Statistics Theory 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. |
| title | Spectra of high-dimensional sparse random geometric graphs |
| topic | Probability Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2507.06556 |