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 \({N\times N}\). 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).
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Belinschi S., Nowak M.A., Speicher R., Tarnowski W.: Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem. J. Phys. A Math. Theor. 50(10), 105204 (2017)
Borodin A., Strahov E.: Averages of characteristic polynomials in random matrix theory. Commun. Pure Appl. Math. 59(2), 161–253 (2006)
Bourgade, P., Dubach, G.: The distribution of overlaps between eigenvectors of Ginibre matrices (2018). arXiv:1801.01219
Burda Z., Grela J., Nowak M.A., Tarnowski W., Warchol P.: Dysonian dynamics of the Ginibre ensemble. Phys. Rev. Lett. 113(10), 104102 (2014)
Burda Z., Grela J., Nowak M.A., Tarnowski W., Warchol P.: Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics. Nucl. Phys. B 897, 421–447 (2015)
Burda Z., Spisak B.J., Vivo P.: Eigenvector statistics of the product of Ginibre matrices. Phys. Rev. E 95(2), 022134 (2017)
Chalker J.T., Mehlig B.: Eigenvector statistics in non-Hermitian random matrix ensembles. Phys. Rev. Lett. 81(16), 3367–3370 (1998)
Desrosiers P., Forrester P.J.: A note on biorthogonal ensembles. J. Approx. Theor. 152(2), 167–187 (2008)
Edelman A., Kostlan E., Shub M.: How many eigenvalues of a real matrix are real?. J. Am. Math. Soc. 7(1), 247–267 (1994)
Edelman A.: The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivar. Anal. 60(2), 203–232 (1997)
Forrester P.J., Nagao T.: Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A Math. Theor. 41(37), 375003 (2008)
Fyodorov Y.V., Sommers H.-J.: Statistics of resonance poles, phaseshifts and time delays in quantum chaotic scattering: random matrix approach for systems with broken time-reversal invariance. J. Math. Phys. 38(4), 1918–1981 (1997)
Fyodorov Y.V., Mehlig B.: Statistics of resonances and nonorthogonal eigenfunctions in a model for single-channel chaotic scattering. Phys. Rev. E 66(4), 045202(R) (2002)
Fyodorov Y.V.: Negative moments of characteristic polynomials of random matrices: Ingham–Siegel integral as an alternative to Hubbard–Stratonovich transformation. Nucl. Phys. B 621(3), 643–674 (2002)
Fyodorov Y.V., Strahov E.: On correlation functions of characteristic polynomials for chiral Gaussian unitary ensemble. Nucl. Phys. B 647 [FS](3), 581–597 (2002)
Fyodorov Y.V., Strahov E.: Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localization on non-compact Kähler manifolds. Nucl. Phys. B 630 [PM](3), 453–491 (2002)
Fyodorov Y.V., Akemann G.: On the supersymmetric partition function in QCD-inspired random matrix models. JETP Lett. 77(8), 438–441 (2003)
Fyodorov Y.V., Sommers H.-J.: Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A Math. Gen. 36(12), 3303–3347 (2003)
Fyodorov Y.V., Strahov E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A Math. Gen. 36(12), 3203–3213 (2003)
Fyodorov Y.V., Khoruzhenko B.A.: On absolute moments of characteristic polynomials of a certain class of complex random matrices. Commun. Math. Phys. 273(3), 561–599 (2007)
Fyodorov, Y.V., Savin, D.V.: Resonance scattering in chaotic systems, chapter 34. In: Akemann, G., Baik, J., Di Francesco, P. (eds.), The Oxford Handbook of Random Matrix Theory, p. 703. Oxford University Press, Oxford (2011)
Fyodorov Y.V., Savin D.V.: Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality. Phys. Rev. Lett. 108(18), 184101 (2012)
Fyodorov Y.V., Nock A.: On random matrix averages involving half-integer powers of GOE characteristic polynomials. J. Stat. Phys. 159(4), 731–751 (2015)
Fyodorov Y.V., Khoruzhenko B.A.: Nonlinear analogue of the May–Wigner instability transition. Proc. Natl. Acad. Sci. USA 113(25), 6827–6832 (2016)
Fyodorov Y.V., Grela J., Strahov E.: On characteristic polynomials for a generalized chiral random matrix ensemble with a source. J. Phys. A Math. Theor. 51(13), 134003 (2018)
Gangulia S., Huhc D., Sompolinsky H.: Memory traces in dynamical systems. Proc. Natl. Acad. Sci. USA 105(48), 18970–18975 (2008)
Goetschy A., Skipetrov S.E.: Non-Hermitian Euclidean random matrix theory. Phys. Rev. E 84(1), 011150 (2011)
Gradshteyn L.S., Ryzhik I.M.: Tables of Integers, Series and Products, 6th ed. Academic Press, New York (2000)
Grela J.: What drives transient behaviour in complex systems?. Phys. Rev. E 96(2), 022316 (2017)
Grela J., Guhr T.: Exact spectral densities of complex noise-plus-structure random matrices. Phys. Rev. E 94(4), 042130 (2016)
Gros J.-B., Kuhl U., Legrand O., Mortessagne F., Richalot E., Savin D.V.: Experimental width shift distribution: a test of nonorthogonality for local and global perturbations. Phys. Rev. Lett. 113(22), 224101 (2014)
Guhr, T.: Supersymmetry. Chapter 7 in The Oxford Handbook of Random Matrix Theory. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford University Press (2011). arXiv:1005.0979 [math-ph]
Janik R.A., Nörenberg W., Nowak M.A., Papp G., Zahed I.: Correlations of eigenvectors for non-Hermitian random-matrix models. Phys. Rev. E 60(3), 2699–2705 (1999)
Khoruzhenko, B.A., Sommers, H.-J.: Non-Hermitian ensembles. Chapter 18 in The Oxford Handbook of Random Matrix Theory. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford University Press (2011)
Kozhan R.: Rank one non-Hermitian perturbations of Hermitian β-ensembles of random matrices. J. Stat. Phys. 168(1), 92–108 (2017)
May R.M.: Will a large complex system be stable?. Nature 238, 413–414 (1972)
Mehlig B., Chalker J.T.: Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles. J. Math. Phys. 41(5), 3233–3256 (2000)
Mehlig B., Santer M.: Universal eigenvector statistics in a quantum scattering ensemble. Phys. Rev. E 63(2), 020105(R) (2001)
Movassagh R.: Eigenvalue attraction. J. Stat. Phys. 162(3), 615–643 (2016)
Neri, I., Metz, F.L.: Eigenvalue outliers of non-Hermitian random matrices with a local tree structure. Phys. Rev. Lett. 117(22), 224101 (2016); Erratum Phys. Rev. Lett. 118, 019901 (2017)
Nowak, M.A., Tarnowski, W.: Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach (2018). arXiv:1801.02526 [math-ph]
Osborn J.C.: Universal results from an alternate random matrix model for QCD with a baryon chemical potential. Phys. Rev. Lett. 93(22), 222001 (2004)
Rotter I.: A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor. 42(15), 153001 (2009)
Savin D.V., Sokolov V.V.: Quantum versus classical decay laws in open chaotic systems. Phys. Rev. E 56(5), R4911–R4913 (1997)
Schomerus H., Frahm K.M., Patra M., Beenakker C.W.J.: Quantum limit of the laser line width in chaotic cavities and statistics of residues of scattering matrix poles. Phys. A 278(3–4), 469–496 (2000)
Seif B., Wettig T., Guhr T.: Spectral correlations of the massive QCD Dirac operator at finite temperature. Nucl. Phys. B 548(1–3), 475–490 (1999)
Trefethen L.N., Trefethen A.E., Reddy S.C., Driscoll T.A.: Hydrodynamic stability without eigenvalues. Science 261(5121), 578–584 (1993)
Trefethen L.N., Embree M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)
Walters M., Starr S.: A note on mixed matrix moments for the complex Ginibre ensemble. J. Math. Phys. 56(1), 013301 (2015)
Wirtz T., Akemann G., Guhr T., Kieburg M., Wegner R.: The smallest eigenvalue distribution in the real WishartLaguerre ensemble with even topology. J. Phys. A Math. Theor. 48(24), 245202 (2015)
Zirnbauer, M.R.: The supersymmetry method of random matrix theory. In: Francoise, J.-P., Naber, G.L., Tsun, T.S. (ed.) Encyclopedia of Mathematical Physics. Academic Press, Oxford (2006). arXiv:math-ph/0404057]
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Fyodorov, Y.V. On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry. Commun. Math. Phys. 363, 579–603 (2018). https://doi.org/10.1007/s00220-018-3163-3
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DOI: https://doi.org/10.1007/s00220-018-3163-3