Abstract
We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.
Similar content being viewed by others
Notes
More precisely, for any smooth, bounded, compactly supported function f and deterministic sequence \((z_N)\) such that \(|z_N|<1-N^{-\frac{1}{2}+\kappa }\) we have \({\mathbb {E}}\left( f({\mathscr {O}}_{11}/(N(1-|z_N|^2)))\mid \lambda _1=z_N\right) \rightarrow {\mathbb {E}}f(\gamma _2^{-1})\).
References
Akemann, G., Tribe, R., Tsareas, A., Zaboronski, O.: On the determinantal structure of conditional overlaps for the complex Ginibre ensemble (2019). arXiv:1903.09016
Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)
Andréief, M.C.: Note sur une relation entre les intégrales définies des produits des fonctions. Mémoires de la société des sciences physiques et naturelles de Bordeaux 2, 1–14 (1883)
Arguin, L.-P., Belius, D., Bourgade, P.: Maximum of the characteristic polynomial of random unitary matrices. Commun. Math. Phys. 349, 703–751 (2017)
Belinschi, S., Nowak, M.A., Speicher, R., Tarnowski, W.: Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem. J. Phys. A 50(10), 105204, 11 (2017)
Bhattacharya, R.N., Ranga Rao, R.: Normal approximation and asymptotic expansions. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1976)
Bolley, F., Chafaï, D., Fontbona, J.: Dynamics of a planar Coulomb gas. Ann. Appl. Probab. 28(5), 3152–3183 (2018)
Bourgade, P., Yau, H.-T.: The eigenvector moment ow and local quantum unique ergodicity. Commun. Math. Phys. 350(1), 231–278 (2017)
Bourgade, P., Yau, H.-T., Yin, J.: The local circular law II: the edge case. Probab. Theory Rel. Fields 159(3–4), 619–660 (2014)
Breuer, J., Duits, M.: The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles. Adv. Math. 265, 441–484 (2014)
Burda, Z., Grela, J., Nowak, M.A., Tarnowski, W., Warchoł, P.: Dysonian dynamics of the Ginibre ensemble. Phys. Rev. Lett. 113, 104102 (2014)
Chalker, J.T., Mehlig, B.: Eigenvector statistics in non-Hermitian random matrix ensembles. Phys. Rev. Lett. 81(16), 3367–3370 (1998)
Chalker, J.T., Mehlig, B.: Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles. J. Math. Phys. 41(5), 3233–3256 (2000)
Crawford, N., Rosenthal, R.: Eigenvector correlations in the complex Ginibre ensemble (2018). arXiv: 1805.08993
Davy, M., Genack, A.Z.: Probing nonorthogonality of eigenfunctions and its impact on transport through open systems. Phys. Rev. Res. 1, 033026 (2019)
Deift, P., Gioev, D.: Random matrix theory: invariant ensembles and universality. Courant Lecture Notes in Mathematics, vol. 18, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2009)
Diaconis, P., Freedman, D.: A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23, 397–423 (1987). (English, with French summary)
Dubach, G.: Powers of Ginibre eigenvalues. Electron. J. Probab. 23, 1–31 (2018)
Dubach, G.: Symmetries of the Quaternionic Ginibre Ensemble. Random Matrices Theory Appl. (to appear, 2019). arXiv:1811.03724
Dubach, G.: On eigenvector statistics in the spherical and truncated unitary ensembles (2019). arXiv:1908.06713
Erdős, L., Krüger, T., Renfrew, D.: Power law decay for systems of randomly coupled differential equations. SIAM J. Math. Anal. 50(3), 3271–3290 (2018)
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.: Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010)
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(2), 579–603 (2018)
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., Savin, D.V.: Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality. Phys. Rev. Lett. 108(18), 184101 (2012)
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)
Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)
Goetschy, A., Skipetrov, S.E.: Non-Hermitian Euclidean random matrix theory. Phys Rev E. 84, 011150 (2011)
Grela, J., Warchoł, P.: Full Dysonian dynamics of the complex Ginibre ensemble. J. Phys. A. Math. Theor. 51(42), 425203 (2018)
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, 224101 (2014)
Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)
Janik, R.A., Noerenberg, 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)
Johnson, N.L., Kotz, S.: Distributions in statistics. Continuous univariate distributions. 2., Houghton Mifflin Co., Boston, Mass (1970)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980). (corrected second edition)
Keller, J.B.: Multiple eigenvalues. Linear Algebra Appl. 429(8–9), 2209–2220 (2008)
Khoruzhenko, B.A., Sommers, H.J.: Non-Hermitian Ensembles. In: Akemann, G., Baik, J., Francesco, P.D. (eds.) The Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011)
Kostlan, E.: On the spectra of Gaussian matrices. Linear Algebra Appl. 162(164), 385–388 (1992). Directions in matrix theory (Auburn, AL, 1990)
Knowles, A., Yin, J.: Eigenvector distribution of Wigner matrices. Probab. Theory Rel. Fields 155(3–4), 543–582 (2013)
Lambert, G.: The law of large numbers for the maximum of the characteristic polynomial of the Ginibre ensemble (2019). arXiv:1902.01983
Lehmann, N., Sommers, H.-J.: Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67(8), 941–944 (1991)
Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212 (2003)
Mehlig, B., Chalker, J.T.: Eigenvector correlations in non-Hermitian random matrix ensembles. Ann. Phys. 7(5–6), 427–436 (1998)
Mehta, M.L.: Random Matrices, 2nd edn. Academic Press Inc, Boston, MA (1991)
Movassagh, R.: Eigenvalue attraction. J. Stat. Phys. 162(3), 615–643 (2016)
Nowak, M.A., Tarnowski, W.: Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach. J. High Energy Phys. 6, 152 (2018)
Overton, M.L., Womersley, R.S.: On minimizing the spectral radius of a nonsymmetric matrix function: optimality conditions and duality theory. SIAM J. Matrix Anal. Appl. 9(4), 473–498 (1988)
Rotter, I.: A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A: Math. Theor. 42, 153001 (2009)
Rudelson, M., Vershynin, R.: Delocalization of eigenvectors of random matrices with independent entries. Duke Math. J. 164(13), 2507–2538 (2015)
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. Physica A 278(3–4), 469–496 (2000)
Tao, T., Vu, V.: Random matrices: universal properties of eigenvectors. Random Matrices Theory Appl. 1(1), 1150001 (2012)
Trefethen, L.N., Embree, M.: Spectra and pseudospectra: the behavior of nonnormai matrices and operators, vol. 1. Princeton University Press, Princeton (2005)
Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stability without eigenvalues. Science 261, 578,584 (1993)
Walters, M., Starr, S.: A note on mixed matrix moments for the complex Ginibre ensemble. J. Math. Phys. 56(1), 013301, 20 (2015)
Webb, C., Wong, M.D.: On the moments of the characteristic polynomial of a Ginibre random matrix. Proc. Lond. Math. Soc. (3) 118(5), 10171056 (2019)
Acknowledgements
The authors thank the referees for particularly precise and pertinent suggestions which helped improving this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the NSF Grant DMS#1513587.
Appendices
Appendix A: Eigenvalues dynamics
This Appendix derives the Dyson-type dynamics for eigenvalues of nonnormal matrices. More precisely, we consider the Ornstein-Uhlenbeck version so that the equilibrium measure is the (real or complex) Ginibre ensemble. These dynamics take a particularly simple form in the case of complex Gaussian addition, where the drift term shows no interaction between eigenvalues: only the correlation of martingale terms is responsible for eigenvalues repulsion.
We also describe natural dynamics with equilibrium measure given by the real Ginibre ensemble. Then, the eigenvalues evolution is more intricate.
It was already noted in [11] that eigenvectors impact the eigenvalues dynamics for nonnormal matrices, and the full dynamics in the complex case have been written down in [30].
1.1 Complex Ginibre dynamics
Let G(0) be a complex matrix of size N, assumed to be diagonalized as \(YGX = \Delta = \mathrm {Diag}(\lambda _1,\dots ,\lambda _N) \), where X, Y are the matrices of the right- and left-eigenvectors of G(0). We also assume that G(0) has simple spectrum, and X, Y invertible. The right eigenvectors \((x_i)\) are the columns of X, and the left-eigenvectors \((y_j)\) are the rows of Y. They are chosen uniquely such that \(XY=I\) and, for any \(1\leqslant k\leqslant N\), \(X_{kk}=1\).
We now consider the complex Dyson-type dynamics: for any \(1\leqslant i,j\leqslant N\),
where the \(B_{ij}\)’s are independent standard complex Brownian motions: \(\sqrt{2}{{\,\mathrm{Re}\,}}(B_{ij})\) and \(\sqrt{2}{{\,\mathrm{Im}\,}}(B_{ij})\) are standard real Brownian motions. One can easily check that G(t) converges to the Ginibre ensemble as \(t\rightarrow \infty \), with normalization (1.1).
In the following, the bracket of two complex martingales M, N is defined by bilinearity: \(\langle M,N\rangle =\langle {{\,\mathrm{Re}\,}}M,{{\,\mathrm{Re}\,}}N\rangle -\langle {{\,\mathrm{Im}\,}}M,{{\,\mathrm{Im}\,}}N\rangle +\mathrm {i}\langle {{\,\mathrm{Re}\,}}M,{{\,\mathrm{Im}\,}}N\rangle +\mathrm {i}\langle {{\,\mathrm{Im}\,}}M,{{\,\mathrm{Re}\,}}N\rangle \).
Proposition A.1
The spectrum \((\lambda _1(t),\dots , \lambda _n(t))\) is a semimartingale satisfying the system of equations
where the martingales \((M_k)_{1\leqslant k\leqslant N}\) have brackets \(\langle M_i,M_j\rangle =0\) and
Remark A.2
As explained below, this equation (in particular the off-diagonal brackets) is coherent with the eigenvalues repulsion observed in (1.2). Contrary to the Hermitian Dyson Brownian motion, all eigenvalues are martingales (up to the Ornstein Uhlenbeck drift term), so that their repulsion is not due to direct mutual interaction, but to correlations between these martingales at the microscopic scale.
For example, assume that G(0) is already at equilibrium. Using physics conventions, for any bulk eigenvalues \(\lambda _1,\lambda _2\) satisfying \(w={{\,\mathrm{O}\,}}(1)\) (remember \(w=\sqrt{N}(\lambda _1-\lambda _2)\)), Proposition A.1 and Theorem 1.4 imply
in the bulk. By considering the real part in this equation and denoting \(\mathrm{d}\lambda _1=\mathrm{d}x_1+\mathrm {i}\mathrm{d}y_1\), \(\mathrm{d}\lambda _2=\mathrm{d}x_2+\mathrm {i}\mathrm{d}y_2\), we have in particular \({\mathbb {E}}(\mathrm{d}x_1\mathrm{d}x_2+\mathrm{d}y_1\mathrm{d}y_2)<0\), and this negative correlation is responsible for repulsion: the eigenvalues tend to move in opposite directions. Moreover, as eigenvalues get closer on the microscopic scale, \(w\rightarrow 0\) and the repulsion gets stronger:
On the other hand, for mesoscopic scale \(N^{-1/2}\ll |\lambda _1-\lambda _2|\), Proposition A.1 and Theorem 1.4 give \( {\mathbb {E}}\left( \mathrm{d}\lambda _1\mathrm{d}\overline{\lambda _2}\right) \sim -\frac{(1-|\lambda _1|^2)}{N^2|\lambda _1-\lambda _2|^4}\mathrm{d}t={{\,\mathrm{o}\,}}(\mathrm{d}t)\), so that increments are uncorrelated for large N.
For a given differential operator \(f\mapsto f'\), we introduce the matrix \(C=X^{-1}X'\). Along the following lemmas, all eigenvalues are assumed to be distinct. In our application, this spectrum simplicity will hold almost surely for any \(t\geqslant 0\) as G(0) has simple spectrum.
Lemma A.3
We have \(X'=XC\) and \(Y'=-CY\).
Proof
The first equality is the definition of C. For the second one, \(XY=I\) gives \(XY'+X'Y=0\), hence \(Y' = - X^{-1} X' Y = -CY.\)\(\square \)
Lemma A.4
The first order perturbation of eigenvalues is given by \(\lambda _k ' = y_k G' x_k\).
Proof
We have \(\Delta ' = (YGX)' = Y'GX + YG'X + YGX' = YG'X + YGXC - C YGX = YG'X + \Delta C - C \Delta = YG'X + [\Delta ,C]\). Therefore \(\lambda _k' = (YG'X)_{kk} + [\Delta ,C]_{kk} = y_k G' x_k\). \(\square \)
Lemma A.5
For any \(i \ne j\), \(C_{ij} = {y_i G' x_j \over \lambda _j - \lambda _i}\).
Proof
For such i, j, \(\Delta '_{ij} = 0\). With the same computation as in the previous lemma, this gives \((YG'X)_{ij} + [\Delta ,C]_{ij}=0\). Thus \((\lambda _i - \lambda _j) C_{ij} = - (YG'X)_{ij} = - y_i G' x_j\), from which the result follows. \(\square \)
Lemma A.6
For any \(1\leqslant k\leqslant N\), \(C_{kk} = - \sum _{l \ne k} X_{kl} {y_l G' x_k \over \lambda _k - \lambda _l}\).
Proof
We use the assumption \( X_{kk}=1\). From this, and the definition of C, we get
As a consequence, \(C_{kk} = - \sum _{l \ne k} X_{kl} C_{lk}\) and we obtain the result thanks to the previous lemma. \(\square \)
From now on the differential operator will be either \(\partial _{{{\,\mathrm{Re}\,}}G_{ab}}\) (\(G' = E_{ab}= \{ \delta _{ia} \delta _{jb} \}_{1\leqslant i,j\leqslant N}\)), or \(\partial _{{{\,\mathrm{Im}\,}}G_{ab}}\), (\(G' = \mathrm {i}E_{ab})\). In both cases, \(G''=0\). We denote \(C^{{{\,\mathrm{Re}\,}}}\) and \(C^{{{\,\mathrm{Im}\,}}}\) accordingly. In particular, for any k and \(i\ne j\) the following holds:
Lemma A.7
We have
Proof
Below is the computation for \( \partial _{{{\,\mathrm{Re}\,}}G_{ab}}X_{ij} \). We use \(X'=XC\) and (A.2):
The case \(\partial _{{{\,\mathrm{Im}\,}}G_{ab}}X_{ij}\) is obtained similarly, as are the formulas for Y. \(\square \)
Lemma A.8
The second order perturbation of eigenvalues is given by
Proof
We compute the perturbation for \( \partial _{{{\,\mathrm{Re}\,}}G_{ab}} \). Differentiating \(\lambda \) a second time gives
Replacing \(X'\) and \(Y'\) with their expressions yields
which concludes the proof, the other cases being similar. \(\square \)
For the proof of Proposition A.1, we need the following elementary lemma.
Lemma A.9
Let \(\tau =\inf \{t\geqslant 0:\exists i\ne j, \lambda _i(t)=\lambda _j(t)\}\). Then \(\tau =\infty \) almost surely.
Proof
The set of matrices G with Jordan form of type
is a submanifold \({\mathcal {M}}_1\) (resp. \({\mathcal {M}}_2\)) of \({\mathbb {C}}^{N^2}\) with complex codimension 1 (resp. 3), see e.g. [36, 47]. Therefore, almost surely, a Brownian motion in \({\mathbb {C}}^{N^2}\) starting from a diagonalizable matrix with simple spectrum will not hit \({\mathcal {M}}_1\) or \({\mathcal {M}}_2\). This concludes the proof. \(\square \)
All derivatives can therefore be calculated, as eigenvalues and eigenvectors are analytic functions of the matrix entries (see [35]).
Proof of Proposition A.1
In our context,the Itô formula will take the following form: for a function f from \({\mathbb {C}}^n\) to \({\mathbb {C}}\) of class \(C^2\), where \(B_t=(B_t^1,\dots ,B_t^n)\) is made of independent standard complex Brownian motions, we have
For any given \(0<{\varepsilon }<\min \{|\lambda _i(0)-\lambda _j(0)|,i\ne j\}\), let
Eigenvalues are smooth functions of the matrix coefficients on the domain \(\cap _{i<j}\{|\lambda _i-\lambda _j|>{\varepsilon }\}\), so that Eq. (A.3) together with Lemmas A.4 and A.8 gives the following equality of stochastic integrals, with substantial cancellations of the drift term:
Taking \({\varepsilon }\rightarrow 0\) in the above equation together with Lemma A.9 yields
The eigenvalues martingales terms are correlated. Their brackets are
This concludes the proof. \(\square \)
1.2 Proof of Corollary 1.6
Let \(A_{t,{\varepsilon }}=\{\sup _{0\leqslant s\leqslant t}|\lambda _1(s)-\lambda _1(0)|<N^{\varepsilon }t^{1/2}\}\). We start by proving that
From Proposition A.1 and Itô’s formula, we have
which is a local martingale. It is an actual martingale because
where in the last equality we used \({\mathbb {E}}({\mathscr {O}}_{11}(s))={{\,\mathrm{O}\,}}(N)\), which follows from (2.11). The estimate (A.7) follows by Doob’s and Markov’s inequalities.
For (1.16), we start with
This implies
Here, we used that \((\mathrm{Re}\int _0^t\overline{e^{\frac{s}{2}}\lambda _1(s)-\lambda _1(0)}e^{\frac{s}{2}}\mathrm{d}M_1(s) )_t\) is an actual martingale, because the expectation of its bracket is
where for the last inequality we used (2.11).
To evaluate the right hand side of (A.11), we would like to change \(\lambda _1(0)\in {\mathscr {B}}\) into \(\lambda _1(s)\in {\mathscr {B}}\). First,
where for the last inequality we used (2.11), again. Moreover, if \(1/p+1/q=1\) with \(p<2\). we have
where we used [24, Theorem 2.3] to obtain that uniformly in the complex plane and in N, \({\mathscr {O}}_{11}/N\) has finite moment of order \(p<2\). Equations (A.11), (A.12) and (A.13) imply
and one concludes the proof of (1.16) with (2.11).
The proof of (1.17) is identical, except that we rely on the off-diagonal bracket \(\mathrm{d}\langle \lambda _1,\bar{\lambda }_2\rangle _s={\mathscr {O}}_{12}(s)\frac{\mathrm{d}s}{N}\), the estimate (1.9), and the elementary inequality
to bound the (first and p-th) moment of \({\mathscr {O}}_{12}\) in the whole complex plane based on those of \({\mathscr {O}}_{11}\), \({\mathscr {O}}_{22}\).
1.3 Real Ginibre dynamics
We now consider G(0) a real matrix of size N, again assumed to be diagonalized as \(YGX = \Delta = \mathrm {Diag}(\lambda _1,\dots ,\lambda _N) \), where X, Y are the matrices of the right- and left-eigenvectors of G(0). We also assume that G(0) has simple spectrum, and X, Y invertible. We keep the same notations for the right eigenvectors \((x_i)\), columns of X, and the left-eigenvectors \((y_j)\), rows of Y. They are again chosen such that \(XY=I\) and, for any \(1\leqslant k\leqslant N\), \(X_{kk}=1\).
In this subsection, the real Dyson-type dynamics are (\(1\leqslant i,j\leqslant N\)),
where the \(B_{ij}\)’s are independent standard Brownian motions. One can easily check that G(t) converges to the real Ginibre ensemble as \(t\rightarrow \infty \).
Note that the real analogue of Lemma A.9 gives weaker repulsion: the set of real matrices with Jordan form of type
is a submanifold \({\mathcal {M}}_1\) of \({\mathbb {R}}^{N^2}\), supported on \(\lambda _{N-1}\in {\mathbb {R}}\), with real codimension 1 (as proved by a straightforward adaptation of [36, Theorem 7]). Denoting \(\tau =\inf \{t\geqslant 0:\exists i\ne j, \lambda _i(t)=\lambda _j(t)\}\), under the dynamics (A.14) for any \(t>0\) we therefore have
so that we can only state the real version of Proposition A.1 up to time \(\tau \). In fact, collisions occur transforming pairs of real eigenvalues into pairs of complex conjugate eigenvalues, a mechanism coherent with the random number of real eigenvalues in the real Ginibre ensemble [22, 41].
The overlaps (1.4) are enough to describe the complex Ginibre dynamics, and so are they for the real Ginibre ensemble, up to the introduction of the following notation: we define \(\bar{i}\in \llbracket 1,N\rrbracket \) through \(\lambda _{\bar{i}}=\overline{\lambda _i}\), i.e. \({\bar{i}}\) is the index of the conjugate eigenvalue to \(\lambda _i\). Note that \({\bar{i}}=i\) if \(\lambda _i\in {\mathbb {R}}\). For real matrices, if \(L_j,R_j\) are eigenvectors associated to \(\lambda _j\), \({\bar{L}}_j,{\bar{R}}_j\) are eigenvectors for \({{\bar{\lambda }}}_j\), so that
Proposition A.10
The spectrum \((\lambda _1(t),\dots , \lambda _n(t))\) evolves according to the following stochastic equations, up to the first collision:
where the martingales \((M_k)_{1\leqslant k\leqslant N}\) have brackets
Note that the real eigenvalues have associated real eigenvectors. For those, \({{\mathscr {O}}}_{k{\bar{l}}}={\mathscr {O}}_{kl}\), and the variation is real: real eigenvalues remain real as long as they do not collide.
Remark A.11
Proposition A.10 is coherent with the attraction between conjugate eigenvalues exhibited in [45]. In fact, if \(\eta ={{\,\mathrm{Im}\,}}(\lambda _k)>0\), the drift interaction term with \(\bar{\lambda }_k\) is \({\mathscr {O}}_{kk}/(\lambda _k-{{\bar{\lambda }}}_k)=-\mathrm {i}{\mathscr {O}}_{kk}/(2\eta )\), so that these eigenvalues attract each other stronger as they approach the real axis.
For the proof, we omit the details and only mention the differences with respect to Proposition A.1. We apply the Itô formula for a \({\mathscr {C}}^2\) function f from \({\mathbb {R}}^n\) to \({\mathbb {C}}\), with argument \(U_t=(U_t^1,\dots ,U_t^n)\) is made of independent Ornstein-Uhlenbeck processes. Together with the perturbation formulas for \(\lambda _k\), Lemmas A.4 and A.8 , we obtain (remember the notation (A.4))
We can take \({\varepsilon }\rightarrow 0\) in the above formulas and the brackets are calculated as follows, concluding the proof:
Appendix B: Normalized eigenvectors
This paper focuses on the condition numbers and off-diagonal overlaps, but the Schur decomposition also easily gives information about other statistics such as the angles between eigenvectors. We include these results for the sake of completeness. We denote the complex angle as
where the phases of \(R_1(1)\) and \(R_2(1)\) can be chosen independent uniform on \([0,2\pi )\). We also define
Proposition B.1
Conditionally on \(\lambda _1=z_1,\lambda _2=z_2\), we have
where \(X \sim {\mathscr {N}}_{{\mathbb {C}}}(0,\frac{1}{2} \mathrm{Id})\).
In particular, for \(\lambda _1, \lambda _2\) at mesoscopic distance, the complex angle converges in distribution to a Dirac mass at 0. Therefore in such a setting eigenvectors strongly tend to be orthogonal: matrices sampled from the Ginibre ensemble are not far from normal, when only considering eigenvectors angles. The limit distribution becomes non trivial in the microscopic scaling \( |\lambda _1 - \lambda _2| \sim N^{-1/2}\), it is the pushforward of a complex Gaussian measure by \(\Phi \).
Proof
From Proposition 2.1 we know that \(R_1^*R_2=R_{T,1}^*R_{T,2}\), \(\Vert R_1\Vert =\Vert R_{T,1}\Vert \) and \(\Vert R_2\Vert =\Vert R_{T,2}\Vert \), where \(R_{T,i}\) (and \(L_{T,i}\)) are the normalized bi-orthogonal bases of right and left eigenvectors for T, defined as (2.2). The first eigenvectors are written \(R_{T,1}=(1,0,\dots ,0)\) and \(R_{T,2}=(a,1,0\dots ,0)\) where \(a=-\bar{b}_2=-\frac{T_{12}}{\lambda _1-\lambda _2}\), with \(T_{12}\) complex Gaussian \({\mathscr {N}}\left( 0,\frac{1}{2N}\mathrm{Id}\right) \), independent of \(\lambda _1\) and \(\lambda _2\). This gives
and concludes the proof. \(\square \)
From Proposition B.1, the distribution of the angle for fixed \(\lambda _1\) and random \(\lambda _2\) can easily be inferred. For example, if \(\lambda _2\) is chosen uniformly among eigenvalues in a macroscopic domain \(\Omega \subset \{|z|<1\}\) with nonempty interior, we obtain the convergence in distribution (\(X_\Omega \) is uniform on \(\Omega \), independent of \({\mathscr {N}}\))
When \(z_1=0\) and \(z_2\) is free, the following gives a more precise distribution, for finite N and in the limit.
Corollary B.2
Conditionally on \(\{ \lambda _1 =0 \}\) we have
where \(U_N\) is an independent random variable uniform on \( \{ 2,\dots ,N\}\), and X has density \( {1 - (1+t) e^{-t} \over t^2} {\mathbf {1}}_{{\mathbb {R}}_+} (t).\)
Proof
From Corollary 5.6, \(N |\lambda _2|^2 \sim \gamma _{U_N}\). Together with Lemma 2.5, this gives
The limiting density then follows from the explicit distribution of \(\beta \) random variables. \(\square \)
Rights and permissions
About this article
Cite this article
Bourgade, P., Dubach, G. The distribution of overlaps between eigenvectors of Ginibre matrices. Probab. Theory Relat. Fields 177, 397–464 (2020). https://doi.org/10.1007/s00440-019-00953-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-019-00953-x