ON THE ALMOST EIGENVECTORS OF RANDOM REGULAR GRAPHS

• 成果类型: Article
• 署名作者: Backhausz, Agnes; Szegedy, Balazs
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Eotvos Lorand University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1294
• 发表日期: 2019
• 页码: 1677-1725
• 关键词: density

摘要

Let d >= 3 be fixed and G be a large random d-regular graph on n vertices. We show that if n is large enough then the entry distribution of every almost eigenvector of G(with entry sum 0 and normalized to have length root n) is close to some Gaussian distribution N(0, sigma) in the weak topology where 0 <= sigma <= 1. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d. processes on the infinite regular tree. In particular, we obtain that if an invariant eigenvector process on the infinite d-regular tree is in the weak closure of factor of i.i.d. processes then it has Gaussian distribution.