Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Abstract

We extend the so-called "single ring theorem"[1], also known as the Haagerup-Larsen theorem[2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix $X$, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of $X$. We show that this object allows to calculate the conditional expectation of the squared eigenvalue condition number. We give examples and we provide cross-check of the analytic prediction by the large scale numerics.

9 pages, 2 figures

References in corpus (22)

• Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance
• Non-Hermitean Random Matrix Theory: method of hermitization
• Eigenvector statistics in non-Hermitian random matrix ensembles
• Non-Gaussian Non-Hermitean Random Matrix Theory: phase transitions and addition formalism
• Spectrum of the Product of Independent Random Gaussian Matrices
• Quantum limit of the laser linewidth in chaotic cavities and statistics of residues of scattering matrix poles
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• Nonhermitean Random Matrix Models
• Quantum Versus Classical Decay Laws in Open Chaotic Systems
• Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
• Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models
• Nonhermitean Random Matrix Models : a Free Random Variable Approach
• On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
• Random regularization of Brown spectral measure
• Large Petermann factor in chaotic cavities with many scattering channels
• Dysonian dynamics of the Ginibre ensemble
• Multiplication law and S transform for non-hermitian random matrices
• "Single Ring Theorem" and the Disk-Annulus Phase Transition
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• New spectral relations between products and powers of isotropic random matrices
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• Spectrum of the Fokker-Planck operator representing diffusion in a random velocity field

Cited by in corpus (19)

• 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
• Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• What drives transient behaviour in complex systems?
• From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
• Full Dysonian dynamics of the complex Ginibre ensemble
• Spectra of large time-lagged correlation matrices from Random Matrix Theory
• Eigenvector statistics of the product of Ginibre matrices
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Spectrum of non-Hermitian deep-Hebbian neural networks
• Universal eigenvector correlations in quaternionic Ginibre ensembles
• Eikonal formulation of large dynamical random matrix models
• Explicit formulas concerning eigenvectors of weakly non-unitary matrices
• Real spectra of large real asymmetric random matrices
• Complete diagrammatics of the single ring theorem
• 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
• Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations