No-gaps delocalization for general random matrices

Abstract

We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its ${\ell }_2$ norm. Our results pertain to a wide class of random matrices, including matrices with independent entries, symmetric and skew-symmetric matrices, as well as some other naturally arising ensembles. The matrices can be real and complex; in the latter case we assume that the real and imaginary parts of the entries are independent.

45 pages

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

Cited by in corpus (2)

• Eigenvectors of random matrices: A survey
• Low-degree factors of random polynomials