Sharp Square Root Bounds for Edge Eigenvector Universality in Sparse Random Regular Graphs

We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow approximately normal distributions. Our main result establishes a Berry-Esseen bound showing convergence to the Gaussian with error $O(\sqrt{d} \cdot N^{-1/6+\varepsilon})$ for degrees $d \leq N^{1/4}$. This bound significantly improves upon previous results that had error terms scaling as $d^3$, and we prove our $\sqrt{d}$ scaling is optimal by establishing a matching lower bound. Our proof combines three techniques: (1) refined concentration inequalities that exploit the specific variance structure of regular graphs, (2) a vector-based analysis of the resolvent that avoids iterative procedures, and (3) a framework combining Stein's method with graph-theoretic tools to control higher-order fluctuations. These results provide sharp constants for eigenvector universality in the transition from sparse to moderately dense graphs.