The distribution of overlaps between eigenvectors of Ginibre matrices

Abstract

We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

47 pages, 2 figures

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Cited by in corpus (22)

• Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond
• Edge Universality for non-Hermitian Random Matrices
• Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis
• Dynamics of a planar Coulomb gas
• Symmetries of the Quaternionic Ginibre Ensemble
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Explicit formulas concerning eigenvectors of weakly non-unitary matrices
• Eikonal formulation of large dynamical random matrix models
• Dynamics of a rank-one perturbation of a Hermitian matrix
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Real spectra of large real asymmetric random matrices
• Eigenvalue processes of Elliptic Ginibre Ensemble and their Overlaps
• Determinantal structure and bulk universality of conditional overlaps in the complex Ginibre ensemble
• Density of small singular values of the shifted real Ginibre ensemble
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations
• Open-system dynamics in local Lindbladians with chaotic spectra
• Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations