Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants

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 $$\sup_{x \in \mathbb{R}} \left|\mathbb{P}\left(\sqrt{N}\langle \mathbf{q}, \mathbf{u}_2 \rangle \leq x\right) - \Phi(x)\right| \leq C_d N^{-1/6+\varepsilon}$$ 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 $K \leq N^{1/10-\delta}$, 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.