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arXiv · 2502.08897

Gaussian Waves and Edge Eigenvectors of Random Regular Graphs

Yukun He·Jiaoyang Huang·Horng-Tzer Yau
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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.

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BibTeXRIS

Yukun He, Jiaoyang Huang, Horng-Tzer Yau. 2025-02-13. Gaussian Waves and Edge Eigenvectors of Random Regular Graphs. https://arxiv.org/abs/2502.08897

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