On delocalization of eigenvectors of random non-Hermitian matrices

• 成果类型: Article
• 署名作者: Lytova, Anna; Tikhomirov, Konstantin
• 署名单位: University of Opole; University System of Georgia; Georgia Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00956-8
• 发表日期: 2020
• 页码: 465-524
• 关键词: smallest singular-value; braesss paradox; UNIVERSALITY; probabilities; statistics; INEQUALITY

摘要

We study delocalization of null vectors and eigenvectors of random matrices with i.i.d entries. Let A be an nxn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n$$\end{document}|I|=m||vI||>= m3/2n3/2logCn||v||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \min \limits _{I\subset [n],\,|I|= m}\Vert \mathbf{{v}}_I\Vert \ge \frac{m<^>{3/2}}{n<^>{3/2}\log <^>Cn}\Vert \mathbf{{v}}\Vert \end{aligned}$$\end{document}for any real eigenvector v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{v}}$$\end{document} and any m is an element of[logCn,n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in [\log <^>C n,n]$$\end{document}, where vI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{v}}_I$$\end{document} denotes the restriction of v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{v}}$$\end{document} to I. Further, when the entries of A are complex, with i.i.d real and imaginary parts, we show that with probability at least 1-e-log2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-e<^>{-\log <^>{2} n}$$\end{document}all eigenvectors of A are delocalized in the sense that minI subset of[n],|I|=m||vI||>= mnlogCn||v||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \min \limits _{I\subset [n],\,|I|= m}\Vert \mathbf{{v}}_I\Vert \ge \frac{m}{n\log <^>Cn}\Vert \mathbf{{v}}\Vert \end{aligned}$$\end{document}for all m is an element of[logCn,n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in [\log <^>C{n},n]$$\end{document}. Comparing with related results, in the range m is an element of[logC ' n,n/logC ' n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in [\log <^>{C'}{n},n/\log <^>{C'}{n}]$$\end{document} in the i.i.d setting and with weaker probability estimates, our lower bounds on ||vI||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \mathbf{{v}}_I\Vert $$\end{document} strengthen an earlier estimate min|I|=m||vI||>= c(m/n)6||v||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \nolimits _{|I|= m}\Vert \mathbf{{v}}_I\Vert \ge c(m/n)<^>6\Vert \mathbf{{v}}\Vert $$\end{document} obtained in Rudelson and Vershynin (Geom Funct Anal 26(6):1716-1776, 2016), and bounds min|I|=m||vI||>= c(m/n)2||v||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \nolimits _{|I|= m}\Vert \mathbf{{v}}_I\Vert \ge c(m/n)<^>2\Vert \mathbf{{v}}\Vert $$\end{document} (in the real setting) and min|I|=m||vI||>= c(m/n)3/2||v||\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \nolimits _{|I|= m}\Vert \mathbf{{v}}_I\Vert \ge c(m/n)<^>{3/2}\Vert \mathbf{{v}}\Vert $$\end{document} (in the complex setting) established in Luh and O'Rourke (Eigenvector delocalization for non-Hermitian random matrices and applications. ). As the case of real and complex Gaussian matrices shows, our bounds are optimal up to the polylogarithmic multiples. We derive stronger estimates without the polylogarithmic error multiples for null vectors of real (n-1)xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-1)\times n$$\end{document} random matrices.