Full Dysonian dynamics of the complex Ginibre ensemble

Abstract

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski-Fokker-Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of $k$-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro- and microscopic limiting laws.

arXiv commentminor typos corrected, some references updated

References in corpus (29)

• Non-Hermitean Random Matrix Theory: method of hermitization
• Eigenvector statistics in non-Hermitian random matrix ensembles
• Correlations of Nearby Levels Induced by a Random Potential
• On the structure of Normal Matrix Model
• Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality
• Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles
• Correlations of Eigenvectors for Non-Hermitian Random-Matrix Models
• On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
• On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
• Large scale correlations in normal and general non-Hermitian matrix ensembles
• 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
• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
• Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• Universal shocks in random matrix theory
• A Note on Mixed Matrix Moments for the Complex Ginibre Ensemble
• What drives transient behaviour in complex systems?
• Dynamics of a planar Coulomb gas
• Spectra of large time-lagged correlation matrices from Random Matrix Theory
• Universal shocks in the Wishart random matrix ensemble - 1
• Eigenvector statistics of the product of Ginibre matrices
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Density of State in a Complex Random Matrix Theory with External Source
• Exact spectral densities of complex noise-plus-structure random matrices
• Diffusion in the space of complex Hermitian matrices - microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial
• Eigenvalue Attraction
• Complete diagrammatics of the single ring theorem

Cited by in corpus (11)

• The distribution of overlaps between eigenvectors of Ginibre matrices
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• Bulk Universality for Complex non-Hermitian Matrices with Independent and Identically Distributed Entries
• Eikonal formulation of large dynamical random matrix models
• Real spectra of large real asymmetric random matrices
• Eigenvalue processes of Elliptic Ginibre Ensemble and their Overlaps
• 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
• Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations