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On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry.
- Source :
-
Communications in Mathematical Physics . Oct2018, Vol. 363 Issue 2, p579-603. 25p. - Publication Year :
- 2018
-
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<inline-graphic></inline-graphic>. First we derive the general finite N expression for the JPD of a real eigenvalue λ<inline-graphic></inline-graphic> 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. <ext-link>arXiv:1801.01219</ext-link>). [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00103616
- Volume :
- 363
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Communications in Mathematical Physics
- Publication Type :
- Academic Journal
- Accession number :
- 131661331
- Full Text :
- https://doi.org/10.1007/s00220-018-3163-3