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

TL;DR

This paper establishes sharp bounds on how eigenvectors of sparse random regular graphs project onto fixed directions, showing convergence to normal distributions with optimal error rates, advancing understanding of eigenvector universality.

Contribution

It provides the first sharp, optimal Berry-Esseen bounds for eigenvector projections in sparse regular graphs, improving previous error estimates significantly.

Findings

Proves convergence of eigenvector projections to Gaussian distributions with error $O(rac{\sqrt{d}}{N^{1/6}})$.

Establishes a matching lower bound, confirming the optimality of the $\sqrt{d}$ scaling.

Introduces a novel combination of concentration inequalities, resolvent analysis, and Stein's method for this problem.

Abstract

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.