Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

Abstract

Backhausz and Szegedy (2019) demonstrated that the almost eigenvectors of random regular graphs converge to Gaussian waves with variance $0\le \ddot{I}{ƒ}^2\le 1$. 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 $\ddot{I}{ƒ}^2=1$. 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.

40 pages. arXiv admin note: text overlap with arXiv:2412.20263