SPARSE REGULAR RANDOM GRAPHS: SPECTRAL DENSITY AND EIGENVECTORS

• 成果类型: Article
• 署名作者: Dumitriu, Ioana; Pal, Soumik
• 署名单位: University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP673
• 发表日期: 2012
• 页码: 2197-2235
• 关键词: local eigenvalue statistics; large random matrices; bulk universality; semicircle law; ensembles; edge; delocalization; fluctuations; CONVERGENCE

摘要

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.