We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex $N \times N$ Ginibre ensemble, conditional on an arbitrary number $k = 1, 2, \dots$ of complex eigenvalues. These objects provide the simplest generalization of the expectations of the diagonal overlap ( $k = 1$ ) and the off-diagonal overlap ( $k = 2$ ) considered originally by Chalker and Mehlig. They also appear naturally in the problem of joint evolution of eigenvectors and eigenvalues for Brownian motions with values in complex matrices studied by the Krakow school. We find that these expectations possess a determinantal structure, where the relevant kernels can be expressed in terms of certain orthogonal polynomials in the complex plane. Moreover, the kernels admit a rather tractable expression for all $N \geq 2$ . This result enables a fairly straightforward calculation of the conditional expectation of the overlap matrix in the local bulk and edge scaling limits as well as the proof of the exact algebraic decay and asymptotic factorization of these expectations in the bulk.

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References

1. G. Akemann and F. Basile, Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry, Nucl. Phys. B 766(1–3) (2007) 150–177, arXiv:math-ph/0606060. Crossref, Google Scholar
2. G. Akemann and G. Vernizzi, Characteristic polynomials of complex random matrix models, Nucl. Phys. B 660(3) (2003) 532–556, arXiv:hep-th/0212051. Crossref, Google Scholar
3. G. Akemann, Y.-P. Förster and M. Kieburg, Eigenvector correlations in quaternionic Ginibre ensembles, in preparation. Google Scholar
4. S. Belinschi, M. A. Nowak, R. Speicher and W. Tarnowski, Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem, J. Phys. A: Math. Theor. 50(10) (2017) 105204, arXiv:1608.04923 [math-ph]. Crossref, Google Scholar
5. F. Benaych-Georges and O. Zeitouni, Eigenvectors of non-normal random matrices, preprint (2018), arXiv:1806.06806 [math.PR]. Google Scholar
6. A. Borodin, Biorthogonal ensembles, Nucl. Phys. B 536(3) (1998) 704–732, arXiv:math/9804027 [math.CA]. Crossref, Google Scholar
7. P. Bourgade and G. Dubach, The distribution of overlaps between eigenvectors of Ginibre matrices, preprint (2018), arXiv:1801.01219 [math.PR]. Google Scholar
8. Z. Burda, J. Grela, M. A. Nowak, W. Tarnowski and P. Warchol, Dysonian dynamics of the Ginibre ensemble, Phys. Rev. Lett. 113(10) (2014) 104102, arXiv:1403.7738 [math-ph]. Crossref, Google Scholar
9. Z. Burda, J. Grela, M. A. Nowak, W. Tarnowski and P. Warchol, Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics, Nucl. Phys. B 897 (2015) 421–447, arXiv:1503.06846 [math-ph]. Crossref, Google Scholar
10. Z. Burda, B. J. Spisak and P. Vivo, Eigenvector statistics of the product of Ginibre matrices, Phys. Rev. E 95(2) (2017) 022134, arXiv:1610.09184 [cond-mat.stat-mech]. Crossref, Google Scholar
11. J. T. Chalker and B. Mehlig, Eigenvector statistics in non-Hermitian random matrix ensembles, Phys. Rev. Lett. 81(16) (1998) 3367–3370, arXiv:cond-mat/9809090 [cond-mat.dis-nn]. Crossref, Google Scholar
12. L. L. Chau and O. Zaboronsky, On the structure of correlation functions in the normal matrix model, Commun. Math. Phys. 196(1) (1998) 203–247, arXiv:hep-th/9711091. Crossref, Google Scholar
13. N. Crawford and R. Rosenthal, Eigenvector correlations in the complex Ginibre ensemble, preprint (2018), arXiv:1805.08993 [math.PR]. Google Scholar
14. G. Dubach, Symmetries of the quaternionic Ginibre ensemble, preprint (2018), arXiv:1811.03724 [math.PR]. Google Scholar
15. P. J. Forrester, Log-Gases and Random Matrices, London Mathematical Society, Vol. 34 (Princeton University Press, 2010). Google Scholar
16. P. J. Forrester, T. Nagao and G. Honner, Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nucl. Phys. B 553(3) (1999) 601–643. Crossref, Google Scholar
17. Y. V. Fyodorov, On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: Combining partial Schur decomposition with supersymmetry, Commun. Math. Phys. 363(2) (2018) 579–603, arXiv:1710.04699 [math-ph]. Crossref, Google Scholar
18. Y. V. Fyodorov, J. Grela and E. Strahov, On characteristic polynomials for a generalized chiral random matrix ensemble with a source, J. Phys. A: Math. Theor. 51(13) (2018) 134003, arXiv:1711.07061 [math-ph]. Crossref, Google Scholar
19. Y. V. Fyodorov and D. V. Savin, Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality, Phys. Rev. Lett. 108(18) (2012) 184101. Crossref, Google Scholar
20. J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6(3) (1965) 440–449. Crossref, Google Scholar
21. J. Grela and P. Warchol, Full Dysonian dynamics of the complex Ginibre ensemble, J. Phys. A: Math. Theor. 51(42) (2018) 425203, arXiv:1804.09740 [math-ph]. Crossref, Google Scholar
22. R. A. Janik, W. Nörenberg, M. A. Nowak, G. Papp and I. Zahed, Correlations of eigenvectors for non-Hermitian random-matrix models, Phys. Rev. E 60(3) (1999) 2699, arXiv:cond-mat/9902314. Crossref, Google Scholar
23. B. A. Khoruzhenko and H.-J. Sommers, Non-Hermitian random matrix ensembles, The Oxford Handbook of Random Matrix Theory, Chap. 18 eds. G. Akemann, J. Baik and P. Di Francesco (Oxford University Press, 2011). Google Scholar
24. D. E. Knuth, Overlapping pfaffians, preprint (1995), arXiv:math/9503234 [math.CO]. Google Scholar
25. K. Luh and S. O’Rourke, Eigenvector delocalization for non-Hermitian random matrices and applications, preprint (2018), arXiv:1810.00489 [math.PR]. Google Scholar
26. R. M. May, Will a large complex system be stable? Nature 238(5364) (1972) 413–414. Crossref, Google Scholar
27. B. Mehlig and J. T. Chalker, Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles, J. Math. Phys. 41(5) (2000) 3233–3256, arXiv:cond-mat/9906279 [cond-mat.dis-nn]. Crossref, Google Scholar
28. M. L. Mehta, Random Matrices, Vol. 142 (Elsevier, 2004). Google Scholar
29. M. A. Nowak and W. Tarnowski, Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: Diagrammatic approach, J. High Energy Phys. 2018(6) (2018) 152, arXiv:1801.02526 [math-ph]. Crossref, Google Scholar
30. F. W. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). Google Scholar
31. M. Rudelson and R. Vershynin, Delocalization of eigenvectors of random matrices with independent entries, Duke Math. J. 164(13) (2015) 2507–2538, arXiv:1306.2887 [math.PR]. Crossref, Google Scholar
32. J. R. Silvester, Determinants of block matrices, The Mathematical Gazette 84(501) (2000) 460–467. Crossref, Google Scholar
33. A. Tsareas, PhD dissertation (in preparation), Warwick University (2020). Google Scholar
34. M. Walters and S. Starr, A note on mixed matrix moments for the complex Ginibre ensemble, J. Math. Phys. 56(1) (2015) 013301, arXiv:1409.4494 [math-ph]. Crossref, Google Scholar
35. K. Zyczkowski and H. J. Sommers, Truncations of random unitary matrices, J. Phys. A: Math. Gen. 33(10) (2000) 2045–2057, arXiv:chao-dyn/9910032. Crossref, Google Scholar