Statistical properties of eigenvectors and eigenvalues of structured random matrices

Truong, K.; Ossipov, A.

Abstract

We study the eigenvalues and the eigenvectors of N X N structured random matrices of the form H = W ~HW+D with diagonal matrices D and W and ~H from the Gaussian Unitary Ensemble. Using the supersymmetry technique we derive general asymptotic expressions for the density of states and the moments of the eigenvectors. We find that the eigenvectors remain ergodic under very general assumptions, but a degree of their ergodicity depends strongly on a particular choice of W and D. For a special case of D = 0 and random W, we show that the eigenvectors can become critical and are characterized by non-trivial fractal dimensions.

Citation

Truong, K., & Ossipov, A. (2018). Statistical properties of eigenvectors and eigenvalues of structured random matrices. Journal of Physics A: Mathematical and Theoretical, 51(6), Article 065001. https://doi.org/10.1088/1751-8121/aaa011

• Journal Article Type: Article
• Acceptance Date: Dec 8, 2017
• Publication Date: Jan 10, 2018
• Deposit Date: Jan 12, 2018
• Publicly Available Date: Jan 11, 2019
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 51
• Issue: 6
• Article Number: 065001
• DOI: https://doi.org/10.1088/1751-8121/aaa011
• Keywords: Random matrix theory; Statistics of eigenvectors; Localization
• Public URL: https://nottingham-repository.worktribe.com/output/904299
• Publisher URL: https://doi.org/10.1088/1751-8121/aaa011
• Additional Information: “This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1751-8121/aaa011