Research output: Contribution to journal › Article

Original language | English |
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Number of pages | 25 |

Journal | Communications in Mathematical Physics |

Early online date | 11 Jun 2018 |

DOIs | |

Publication status | E-pub ahead of print - 11 Jun 2018 |

**On Statistics of Bi-Orthogonal Eigenvectors_FYODOROV_Firstonline11June2018_GOLD CP (CC BY)**On_Statistics_of_Bi_Orthogonal_Eigenvectors_FYODOROV_Firstonline11June2018_GOLD_CP_CC_BY_.pdf, 597 KB, application/pdf

11/07/2018

Proof

CC BY

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 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.

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