Noise sensitivity of the top eigenvector of a Wigner matrix

• 成果类型: Article
• 署名作者: Bordenave, Charles; Lugosi, Gabor; Zhivotovskiy, Nikita
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Aix-Marseille Universite; Aix-Marseille Universite; Pompeu Fabra University; ICREA; Alphabet Inc.; Google Incorporated
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00970-1
• 发表日期: 2020
• 页码: 1103-1135
• 关键词: universality

摘要

We investigate the noise sensitivity of the top eigenvector of a Wigner matrix in the following sense. Let v be the top eigenvector of an NxN Wigner matrix. Suppose that k randomly chosen entries of the matrix are resampled, resulting in another realization of the Wigner matrix with top eigenvector v[k]. We prove that, with high probability, when kMUCH LESS-THANN5/3-o(1), then v and v[k] are almost collinear and when k >> N5/3, then v[k] is almost orthogonal to v.