Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

TL;DR

This paper provides an alternative proof that edge eigenvectors of random regular graphs converge to Gaussian waves with variance 1, and shows their asymptotic independence from eigenvalues, using a new Green's function approach.

Contribution

It introduces a novel framework linking Green's function convergence to eigenvector behavior, specifically proving variance equals 1 for edge eigenvectors.

Findings

Edge eigenvectors converge to Gaussian waves with variance 1

Eigenvalues and eigenvectors are asymptotically independent

New Green's function approach links convergence of imaginary part to eigenvector behavior

Abstract

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance $0\leq \sigma^2\leq 1$. In this paper, we present an alternative proof of this result for the edge eigenvectors of random regular graphs, establishing that the variance must be $\sigma^2=1$. Furthermore, we show that the eigenvalues and eigenvectors are asymptotically independent. Our approach introduces a simple framework linking the weak convergence of the imaginary part of the Green's function to the convergence of eigenvectors, which may be of independent interest.