On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry

Abstract

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 × N . First we derive the general finite N expression for the JPD of a real eigenvalue λ 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 nonorthogonality 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 (Phys Rev Lett 81(16):3367-3370, 1998), 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 (The distribution of overlaps between eigenvectors of Ginibre matrices, 2018. arXiv:1801.01219).

On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

“…The same mechanism applies to the overlaps in the quaternionic Ginibre ensemble [20,21]. In the real ensemble [23] the Laplace transformed joint density of overlap and conditional eigenvalue is given by an averaged ratio of characteristic polynomials, thus only depending on the eigenvalues too.…”

“…The quaternionic Ginibre ensemble appeared more recently from a probabilistic angle [19,20] as well as for finite-N in [21], using the heuristic tools of [22]. An entirely different approach uses supersymmetry [23] or orthogonal polynomials [24], expressing the relevant quantities in terms of expectation values of characteristic polynomials. This includes also eigenvectors of real eigenvalues of the real Ginibre ensemble [23,25].…”

“…An entirely different approach uses supersymmetry [23] or orthogonal polynomials [24], expressing the relevant quantities in terms of expectation values of characteristic polynomials. This includes also eigenvectors of real eigenvalues of the real Ginibre ensemble [23,25].…”

Universal eigenvector correlations in quaternionic Ginibre ensembles

“…For the diagonal overlaps O ii the term self-overlap is also used [26], for i = j the O ij are called off-diagonal overlaps. As in [41] the components O ij are invariant under the transformation…”

“…Rigorous results have been obtained for the distribution of diagonal and off-diagonal overlaps and their correlations for the complex Ginibre ensemble (GinUE) in [10] using probabilistic, and in [26,27] using supersymmetric and orthogonal polynomial techniques, respectively. The latter also included partial results on the real eigenvalues of the real Ginibre ensemble (GinOE).…”