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

Yan V Fyodorov

doi:10.1007/s00220-018-3163-3

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 {Ntimes 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 (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).

Highlights

Let x be a N -component column vector, real or complex
The real diagonal entries Oaa are known in the literature on numerical analysis as eigenvalue condition numbers and characterize sensitivity of eigenvalues λa to perturbation of entries of G, see e.g. [48]
Note that the left and right eigenvectors of real-valued matrices corresponding to real eigenvalues λ can be chosen real as well

Summary

Introduction

Let x be a N -component column vector, real or complex. We will use xT = (x1, . . . , xN )to denote the corresponding transposed row vector (and similar notation for matrices), and x∗ = (x1, . . . , x N ) for the Hermitian conjugate, with bar standing for complex conjugation. For the case of complex Ginibre ensemble Gin2 the corresponding joint probability density P(c)(t, z) of the non-orthogonality variable t = Oz −1 and the associated complex eigenvalue z can be found in explicit form for finite N as well, but turns out to be given by a much more cumbersome expression in comparison with the real Ginibre case.

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Conclusion