Localization and delocalization of eigenvectors for heavy-tailed random matrices

• 成果类型: Article
• 署名作者: Bordenave, Charles; Guionnet, Alice
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Ecole Normale Superieure de Lyon (ENS de LYON)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0473-9
• 发表日期: 2013
• 页码: 885-953
• 关键词: semicircle law; UNIVERSALITY; spectrum; statistics

摘要

Consider an Hermitian random matrix with, above the diagonal, independent entries with -stable symmetric distribution and . We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as goes to infinity. When and , we give vanishing bounds on the -norm of the eigenvectors normalized to have unit -norm. On the contrary, when , we prove that these eigenvectors are localized.