Optimal Delocalization for Non--Hermitian Eigenvectors

2025/09/18 by Giorgio Cipolloni, Cipolloni, Giorgio, Landon, Benjamin
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Probability (math.PR) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2509.15189

openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an optimal order delocalization estimate for the eigenvectors of general N × N non-Hermitian matrices X: ‖ \bf v ‖_∞ ≤ C √((log N)/(N)) with very high probability, for any right or left eigenvector \bf v of X. This improves upon the previous tightest bound of Rudelson and Vershynin [arXiv:1306.2887] of O( ( log N)^9/2N^-1/2), and holds under weaker assumptions on the tail of the matrix elements. In addition to the coordinate basis, our bound holds for the ℓ^∞ norm in any deterministic orthonormal basis. Our result is proven via a dynamical method, by studying the flow of the resolvent of the Hermitization of X and proving local laws on short scales.

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Related

• Eigenvector Delocalization for Non-Hermitian Random Matrices and Applications
• On delocalization of eigenvectors of random non-Hermitian matrices
• Optimal Delocalization for Generalized Wigner Matrices
• Localization and delocalization of eigenvectors for heavy-tailed random matrices
• Delocalization of eigenvectors of random matrices. Lecture notes