Eigenvector statistics of the product of Ginibre matrices

Abstract

We develop a method to calculate left-right eigenvector correlations of the product of $m$ independent $N\times N$ complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small $m$ and $N$. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula $O=1+(m/2)(N-1)$ and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as $N\to \infty$ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

arXiv comment25 pag., 8 fig - Typos fixed, minor improvements in presentation

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• Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Full Dysonian dynamics of the complex Ginibre ensemble
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Universal eigenvector correlations in quaternionic Ginibre ensembles
• Spectral statistics for the difference of two Wishart matrices
• 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