On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry

Abstract

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive the general finite $N$ expression for the JPD of a real eigenvalue $\lambda$ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue $z$ and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its 'bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we provide the 'edge' scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach \cite{BourgadeDubach}.

published version

References in corpus (10)

• Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations
• Dysonian dynamics of the Ginibre ensemble
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• On absolute moments of characteristic polynomials of a certain class of complex random matrices
• Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• What drives transient behaviour in complex systems?
• On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials
• On characteristic polynomials for a generalized chiral random matrix ensemble with a source
• Eigenvector statistics of the product of Ginibre matrices

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