Eigenvector statistics in non-Hermitian random matrix ensembles

Abstract

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random variables. Calculating ensemble averages based on the quantity $<L_{\alpha }|L_{\beta }><R_{\beta }|R_{\alpha }>$, where $<L_{\alpha }|$ and $|R_{\beta }>$ are left and right eigenvectors of J, we show for large N that eigenvectors associated with a pair of eigenvalues are highly correlated if the two eigenvalues lie close in the complex plane. We examine consequences of these correlations that are likely to be important in physical applications.

4 pages, no figures

References in corpus (1)

• Random Dirac Fermions and Non-Hermitian Quantum Mechanics

Cited by in corpus (109)

• Non-Hermitian Physics
• Random Matrices close to Hermitian or unitary: overview of methods and results
• Localization transition, spectrum structure and winding numbers for one-dimensional non-Hermitian quasicrystals
• Exact mobility edges, $\mathcal{P}\mathcal{T}$-symmetry breaking and skin effect in one-dimensional non-Hermitian quasicrystals
• Non-Hermitian mobility edges in one-dimensional quasicrystals with parity-time symmetry
• Non-Hermitian disorder in two-dimensional optical lattices
• Universality classes of non-Hermitian random matrices
• Universal spectra of random Lindblad operators
• Field quantization for open optical cavities
• Nonorthogonal pairs of copropagating optical modes in deformed microdisk cavities
• Quantum limit of the laser linewidth in chaotic cavities and statistics of residues of scattering matrix poles
• Nonreciprocal response theory of nonhermitian mechanical metamaterials: response phase transition from the skin effect of zero modes
• Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks
• Anomalous Skin Effects in Disordered Systems with a Single non-Hermitian Impurity
• Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
• Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice
• Non-Hermitian Euclidean random matrix theory
• Eigenvalue distributions of large Euclidean random matrices for waves in random media
• Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
• Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond
• Field quantization for chaotic resonators with overlapping modes
• Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem
• The distribution of overlaps between eigenvectors of Ginibre matrices
• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Dimerization induced mobility edges and multiple reentrant localization transitions in non-Hermitian quasicrystals
• Non-Hermitian Floquet Topological Matter -- A Review
• Large Petermann factor in chaotic cavities with many scattering channels
• Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations
• Dysonian dynamics of the Ginibre ensemble
• Dynamical evolution in a one-dimensional incommensurate lattice with $\mathcal{P}\mathcal{T}$ symmetry
• Resonances in open quantum maps
• Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase
• Petermann factors and phase rigidities near exceptional points
• Quantum-limited linewidth of a chaotic laser cavity
• Random generators of Markovian evolution: A quantum-classical transition by superdecoherence
• Spectral radius of random matrices with independent entries
• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
• "Single Ring Theorem" and the Disk-Annulus Phase Transition
• Driving-induced multiple $\mathcal{P}\mathcal{T}$-symmetry breaking transitions and reentrant localization transitions in non-Hermitian Floquet quasicrystals
• Multiplication law and S transform for non-hermitian random matrices
• Network structural origin of instabilities in large complex systems
• Selectively exciting quasi-normal modes in open disordered systems
• Non-Hermitian quasicrystal in dimerized lattices
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Inhomogeneous losses and complexness of wave functions in chaotic cavities
• Non-Abelian generalization of non-Hermitian quasicrystal: PT-symmetry breaking, localization, entanglement and topological transitions
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Delocalization in an open one-dimensional chain in an imaginary vector potential
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• A model for chaotic dielectric microresonators
• Non-Hermiticity and Universality
• Eigenvalue spectra of large correlated random matrices
• What drives transient behaviour in complex systems?
• Universal hypotrochoidic law for random matrices with cyclic correlations
• Robust localized zero-energy modes from locally embedded PT-symmetric defects
• Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis
• The high temperature crossover for general 2D Coulomb gases
• Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble
• From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
• Formation and interaction of resonance chains in the open 3-disk system
• Universal eigenvector statistics in a quantum scattering ensemble
• Full Dysonian dynamics of the complex Ginibre ensemble
• Impact of non-Hemiticity on modal strength and correlation in transmission through random open cavities
• Spectra of large time-lagged correlation matrices from Random Matrix Theory
• Nonorthogonality constraints in open quantum and wave systems
• Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms
• Eigenvector statistics of the product of Ginibre matrices
• Non-Hermitian Fabry-Perot Resonances in a PT-symmetric system
• Exact complex mobility edges and flagellate spectra for non-Hermitian quasicrystals with exponential hoppings
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes
• Summing free unitary random matrices
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Power law decay for systems of randomly coupled differential equations
• Spectrum of non-Hermitian deep-Hebbian neural networks
• Eigenvector Correlations Across the Localisation Transition in non-Hermitian Power-Law Banded Random Matrices
• Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices
• Universal eigenvector correlations in quaternionic Ginibre ensembles
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Optimal Fluctuations and Tail States of non-Hermitian Operators
• Explicit formulas concerning eigenvectors of weakly non-unitary matrices
• Eikonal formulation of large dynamical random matrix models
• Geometric contribution to adiabatic amplification in non-Hermitian systems
• Probing eigenfunction nonorthogonality by parametric shifts of resonance widths
• Infinite stable Boltzmann planar maps are subdiffusive
• Generalized Petermann factor of non-Hermitian systems at exceptional points
• Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links
• Eigenvalue processes of Elliptic Ginibre Ensemble and their Overlaps
• Complete diagrammatics of the single ring theorem
• The decay of photoexcited quantum systems: a description within the statistical scattering model
• Quaternionic R transform and non-hermitian random matrices
• Density of small singular values of the shifted real Ginibre ensemble
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Optimal Fluctuations and Tail States of non-Hermitian Operators
• Searching for long time scales without fine tuning
• Invariant Eigen-Structure in Complex-Valued Quantum Mechanics
• Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise
• Theory of Localization in a Disordered Ring in a Non-Hermitian Field
• Correspondence principle, dissipation, and Ginibre ensemble
• Concentration of Measure Techniques and Applications
• Open-system dynamics in local Lindbladians with chaotic spectra
• Distance between exceptional points and diabolic points and its implication for the response strength of non-Hermitian systems
• Zero-temperature dynamics of the spherical model with non-reciprocal interactions
• Non-normality and dissipation in Markovian quantum dynamics: Implications for quantum simulation
• Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations