Extremal eigenvectors of sparse random matrices

• 成果类型: Article; Early Access
• 署名作者: He, Yukun; Huang, Jiaoyang; Wang, Chen
• 署名单位: Fudan University; University of Pennsylvania; City University of Hong Kong
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01472-2
• 发表日期: 2026-03-06
• 关键词: eigenvalue statistics; spectral statistics; fluctuations

摘要

We consider a class of sparse random matrices, which includes the adjacency matrix of the Erd & odblac;s-R & eacute;nyi graph G(N,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{G}(N,p)$$\end{document}. For N-1+o(1)<= p <= 1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N<^>{-1+o(1)}\leqslant p\leqslant 1/2$$\end{document}, we show that the non-trivial edge eigenvectors are asymptotically jointly normal. The main ingredient of the proof is an algorithm that directly computes the joint eigenvector distributions, without comparisons with GOE. The method is applicable in general. As an illustration, we also use it to prove the normal fluctuation in quantum ergodicity at the edge for Wigner matrices. Another ingredient of the proof is the isotropic local law for sparse matrices, which at the same time improves several existing results.