On delocalization of eigenvectors of random non-Hermitian matrices

• Published in: Probability theory and related fields 2020-06, Vol.177 (1-2), p.465-524
• Main Authors: Lytova, Anna, Tikhomirov, Konstantin
• Format: Article
• Language: English
• Subjects: Economics; Eigenvectors; Finance; Insurance; Lower bounds; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Matrix algebra; Matrix methods; Operations Research/Decision Theory; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Theoretical
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-019-00956-8

We study delocalization of null vectors and eigenvectors of random matrices with i.i.d entries. Let A be an n × n random matrix with i.i.d real subgaussian entries of zero mean and unit variance. We show that with probability at least 1 - e - log 2 n min I ⊂ [ n ] , | I | = m ‖ v I ‖ ≥ m 3 / 2 n 3 / 2 log C n ‖ v ‖ for any real eigenvector v and any m ∈ [ log C n , n ] , where v I denotes the restriction of v 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 - log 2 n all eigenvectors of A are delocalized in the sense that min I ⊂ [ n ] , | I | = m ‖ v I ‖ ≥ m n log C n ‖ v ‖ for all m ∈ [ log C n , n ] . Comparing with related results, in the range m ∈ [ log C ′ n , n / log C ′ n ] in the i.i.d setting and with weaker probability estimates, our lower bounds on ‖ v I ‖ strengthen an earlier estimate min | I | = m ‖ v I ‖ ≥ c ( m / n ) 6 ‖ v ‖ obtained in Rudelson and Vershynin (Geom Funct Anal 26(6):1716–1776, 2016), and bounds min | I | = m ‖ v I ‖ ≥ c ( m / n ) 2 ‖ v ‖ (in the real setting) and min | I | = m ‖ v I ‖ ≥ c ( m / n ) 3 / 2 ‖ v ‖ (in the complex setting) established in Luh and O’Rourke (Eigenvector delocalization for non-Hermitian random matrices and applications. arXiv:1810.00489 ). 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 ) × n random matrices.