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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.12502 |
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| _version_ | 1866908453706924032 |
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| author | Nagel, Leonhard |
| author_facet | Nagel, Leonhard |
| contents | We establish the first quantitative Berry-Esseen bounds for edge eigenvector statistics in random regular graphs. For any $d$-regular graph on $N$ vertices with fixed $d \geq 3$ and deterministic unit vector $\mathbf{q} \perp \mathbf{e}$, we prove that the normalized overlap $\sqrt{N}\langle \mathbf{q}, \mathbf{u}_2 \rangle$ satisfies \[ \sup_{x \in \mathbb{R}} \left|\mathbb{P}\left(\sqrt{N}\langle \mathbf{q}, \mathbf{u}_2 \rangle \leq x\right) - Φ(x)\right| \leq C_d N^{-1/6+\varepsilon} \] where $\mathbf{u}_2$ is the second eigenvector and $C_d \leq \tilde{C}d^3\varepsilon^{-10}$ for an absolute constant $\tilde{C}$. This provides the first explicit convergence rate for the recent edge eigenvector universality results of He, Huang, and Yau \cite{HHY25}.
Our proof introduces a single-scale comparison method using constrained Dyson Brownian motion that preserves the degree constraint $\tilde{H}_t\mathbf{e} = 0$ throughout the evolution. The key technical innovation is a sharp edge isotropic local law with explicit constant $C(d,\varepsilon) \leq \tilde{C}d\varepsilon^{-5}$, enabling precise control of eigenvector overlap dynamics.
At the critical time $t_* = N^{-1/3+\varepsilon}$, we perform a fourth-order cumulant comparison with constrained GOE, achieving optimal error bounds through a single comparison rather than the traditional multi-scale approach. We extend our results to joint universality for the top $K$ edge eigenvectors with $K \leq N^{1/10-δ}$, showing they converge to independent Gaussians. Through analysis of eigenvalue spacing barriers, critical time scales, and comparison across multiple proof methods, we provide evidence that the $N^{-1/6}$ rate is optimal for sparse regular graphs. All constants are tracked explicitly throughout, enabling finite-size applications in spectral algorithms and network analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants Nagel, Leonhard Probability Discrete Mathematics Combinatorics Spectral Theory 60B20, 15B52, 05C80, 60F05 G.2.2; G.3; F.2.1; G.1.3 We establish the first quantitative Berry-Esseen bounds for edge eigenvector statistics in random regular graphs. For any $d$-regular graph on $N$ vertices with fixed $d \geq 3$ and deterministic unit vector $\mathbf{q} \perp \mathbf{e}$, we prove that the normalized overlap $\sqrt{N}\langle \mathbf{q}, \mathbf{u}_2 \rangle$ satisfies \[ \sup_{x \in \mathbb{R}} \left|\mathbb{P}\left(\sqrt{N}\langle \mathbf{q}, \mathbf{u}_2 \rangle \leq x\right) - Φ(x)\right| \leq C_d N^{-1/6+\varepsilon} \] where $\mathbf{u}_2$ is the second eigenvector and $C_d \leq \tilde{C}d^3\varepsilon^{-10}$ for an absolute constant $\tilde{C}$. This provides the first explicit convergence rate for the recent edge eigenvector universality results of He, Huang, and Yau \cite{HHY25}. Our proof introduces a single-scale comparison method using constrained Dyson Brownian motion that preserves the degree constraint $\tilde{H}_t\mathbf{e} = 0$ throughout the evolution. The key technical innovation is a sharp edge isotropic local law with explicit constant $C(d,\varepsilon) \leq \tilde{C}d\varepsilon^{-5}$, enabling precise control of eigenvector overlap dynamics. At the critical time $t_* = N^{-1/3+\varepsilon}$, we perform a fourth-order cumulant comparison with constrained GOE, achieving optimal error bounds through a single comparison rather than the traditional multi-scale approach. We extend our results to joint universality for the top $K$ edge eigenvectors with $K \leq N^{1/10-δ}$, showing they converge to independent Gaussians. Through analysis of eigenvalue spacing barriers, critical time scales, and comparison across multiple proof methods, we provide evidence that the $N^{-1/6}$ rate is optimal for sparse regular graphs. All constants are tracked explicitly throughout, enabling finite-size applications in spectral algorithms and network analysis. |
| title | Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants |
| topic | Probability Discrete Mathematics Combinatorics Spectral Theory 60B20, 15B52, 05C80, 60F05 G.2.2; G.3; F.2.1; G.1.3 |
| url | https://arxiv.org/abs/2507.12502 |