Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity

2024/02/14 by Mark J. Crumpton, Crumpton, Mark J., Yan V. Fyodorov +3 · 2 citations
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.09296

openalex publication_date 2024/02/14 · openalex created_date 2024/02/18 · openalex updated_date 2026/07/28

Abstract

We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in N× N non-Hermitian random Gaussian matrices. In well known works by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as N→ ∞. In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles characterized by correlations between off-diagonal entries controlled by a parameter τ∈[0,1], with τ=1 corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite N, for any eigenvalue off the real axis. We further investigate several scaling regimes as N→ ∞, both in the limit of strong non-Hermiticity keeping a fixed τ∈[0,1) and in the weak non-Hermiticity limit, with τ approaching unity in such a way that N(1-τ) remains finite.

Cited by

• Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices
• Zero-temperature dynamics of the spherical model with non-reciprocal interactions

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