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 3 and deterministic unit vector q e, we prove that the normalized overlap N q, u₂ satisfies \ ₗ ₑ |P (N q, u₂ x) - Φ (x) | Cd N^-1/6+ \ where u₂ is the second eigenvector and Cd Cd³^-10 for an absolute constant C. This provides the first explicit convergence rate for the recent edge eigenvector universality results of He, Huang, and Yau HHY25. Our proof introduces a single-scale comparison method using constrained Dyson Brownian motion that preserves the degree constraint Hₜe = 0 throughout the evolution. The key technical innovation is a sharp edge isotropic local law with explicit constant C (d, ) Cd^-5, enabling precise control of eigenvector overlap dynamics. At the critical time t_* = N^-1/3+, 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 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.
Leonhard Nagel (2025) studied this question.