On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry. (English) Zbl 1448.60022
Summary: 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 J. T. Chalker and B. Mehlig [“Eigenvector statistics in non-Hermitian random matrix ensembles”, Phys. Rev. Lett. 81, No. 16, 3367–3370 (1998; doi:10.1103/physrevlett.81.3367)], 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 P. Bourgade and G. Dubach [Probab. Theory Relat. Fields 177, No. 1–2, 397–464 (2020; Zbl 1451.60015)].
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