Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Outliers in the Single Ring Theorem

Abstract

This text is about spiked models of non-Hermitian random matrices. More specifically, we consider matrices of the type \({\mathbf {A}}+{\mathbf {P}}\), where the rank of \({\mathbf {P}}\) stays bounded as the dimension goes to infinity and where the matrix \({\mathbf {A}}\) is a non-Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if \({\mathbf {P}}\) has some eigenvalues out of the maximal circle of the single ring, then \({\mathbf {A}}+{\mathbf {P}}\) has some eigenvalues (called outliers) in the neighborhood of those of \({\mathbf {P}}\), which is not the case for the eigenvalues of \({\mathbf {P}}\) in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of \({\mathbf {A}}\) around the eigenvalues of \({\mathbf {P}}\) and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such kind of fluctuations had already been shown for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of \({\mathbf {P}}\)) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in (Ann Probab 42:1980–2031, 2014) in the Hermitian case, but only for non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a result by Tao proved specifically for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    Recall that any matrix \({\mathbf {M}}\) in the set \( {\fancyscript{M}}_n({\mathbb {C}})\) of \(n\times n\) complex matrices is similar to a square block diagonal matrix

    $$\begin{aligned} \left( \begin{array}{lllll} {\mathbf {R}}_{p_1}(\theta _1) &{} &{} &{} (0) \\ &{} {\mathbf {R}}_{p_2}(\theta _2) &{} &{} \\ &{} &{} \ddots &{} \\ (0)&{} &{} &{} {\mathbf {R}}_{p_r}(\theta _r) \\ \end{array} \right) \quad \text { where } \quad {\mathbf {R}}_{p}(\theta ) = \left( \begin{array}{lllll} \theta &{} 1 &{} &{} (0) \\ &{} \ddots &{} \ddots &{} \\ &{} &{} \ddots &{} 1 \\ (0)&{} &{} &{} \theta \\ \end{array} \right) \in {\fancyscript{M}}_p({\mathbb {C}}), \end{aligned}$$

    which is called the Jordan Canonical Form of \({\mathbf {M}}\), unique up to the order of the diagonal blocks [21], Chap. 3].

  2. 2.

    To sort out misunderstandings: we call the multiplicity of an eigenvalue its order as a root of the characteristic polynomial, which is greater than or equal to the dimension of the associated eigenspace.

  3. 3.

    For any \(\sigma >0\), \({\fancyscript{N}}_{{\mathbb {C}}}(0, \sigma ^2)\) denotes the centered Gaussian law on \({\mathbb {C}}\) with covariance \(\frac{1}{2}\begin{pmatrix}\sigma ^2&{}0\\ 0&{}\sigma ^2\end{pmatrix}\).

  4. 4.

    Let us recall that what is here called the multiplicity of an eigenvalue its order as a root of the characteristic polynomial, which is not smaller than the dimension of the associated eigenspace.

  5. 5.

    There is actually another assumption in the Single Ring Theorem [18], but Rudelson and Vershynin recently showed in [28] that it was unnecessary. In [4], Basak Dembo also weakened the hypotheses (roughly allowing Hypothesis 3 not to hold on a small enough set, so that \(\nu \) is allowed to have some atoms). As it follows from the recent preprint [6] that the convergence of the extreme eigenvalues first established in [19] also works in this case, we could harmlessly weaken our hypotheses down to the ones of [4].

  6. 6.

    Recall the \(\beta _{i,j}\) is the number of blocks \({\mathbf {R}}_{p_{i,j}}(\theta _i)\) in the JCF of \({\mathbf {P}}\).

  7. 7.

    Recall that the notion of multiset has been defined before Lemma 5.8: a multiset is roughly a collection of elements with possible repetitions (as in tuples) but where the order of appearance is insignificant (contrarily to tuples). For example,

    $$\begin{aligned} \left\{ 1,2,2,3 \right\} _m =\left\{ 3,2,1,2 \right\} _m \ \ne \ \left\{ 1,2,3 \right\} _m. \end{aligned}$$

References

  1. 1.

    Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge studies in advanced mathematics, 118 (2009)

  2. 2.

    Bai, Z.D., Silverstein, J.W.: Spectral analysis of large dimensional random matrices, 2nd edn. Springer, New York (2009)

  3. 3.

    Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33(5), 1643–1697 (2005)

  4. 4.

    Basak, A., Dembo, A.: Limiting spectral distribution of sums of unitary and orthogonal matrices. Electron. Commun. Probab. 18(69), 19 (2013)

  5. 5.

    Beardon, A.: Complex Analysis: the Winding Number principle in analysis and topology. Wiley, New York (1979)

  6. 6.

    Benaych-Georges, F.: Exponential bounds for the support convergence in the Single Ring Theorem. J. Funct. Anal. 268, 3492–3507 (2015)

  7. 7.

    Benaych-Georges, F., Guionnet, A., Maida M.: Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Electron. J. Prob. 16 (2011) (Paper no. 60, 1621–1662)

  8. 8.

    Benaych-Georges, F., Guionnet, A., Maida, M.: Large deviations of the extreme eigenvalues of random deformations of matrices. Probab. Theory Related Fields 154(3), 703–751 (2012)

  9. 9.

    Benaych-Georges, F., Rao, R.N.: The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227(1), 494–521 (2011)

  10. 10.

    Benaych-Georges, F., Rao, R.N.: The singular values and vectors of low rank perturbations of large rectangular random matrices. J. Multivariate Anal. 111, 120–135 (2012)

  11. 11.

    Bordenave, C., Capitaine, M.: Outlier eigenvalues for deformed i.i.d. random matrices. arXiv:1403.6001

  12. 12.

    Bordenave, C., Chafaï, D.: Around the circular law. Probab. Surv. 9, 1–89 (2012)

  13. 13.

    Capitaine, M., Donati-Martin, C., Féral, D.: The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Ann. Probab. 37, 1–47 (2009)

  14. 14.

    Capitaine, M., Donati-Martin, C., Féral, D.: Central limit theorems for eigenvalues of deformations of Wigner matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48(1), 107–133 (2012)

  15. 15.

    Capitaine, M., Donati-Martin, C., Féral, D., Février, M.: Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Electron. J. Prob. 16, 1750–1792 (2011)

  16. 16.

    Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264(3), 773–795 (2006)

  17. 17.

    Féral, D., Péché, S.: The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272, 185–228 (2007)

  18. 18.

    Guionnet, A., Krishnapur, M., Zeitouni, O.: The Single Ring Theorem. Ann. Math. 174(2), 1189–1217 (2011)

  19. 19.

    Guionnet, A., Zeitouni, O.: Support convergence in the Single Ring Theorem. Probab. Theory Related Fields 154(3–4), 661–675 (2012)

  20. 20.

    Hiai, F., Petz D.: The semicircle law, free random variables, and entropy. Amer. Math. Soc., Math. Surv. Monogr. 77 (2000)

  21. 21.

    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press. ISBN: 978-0-521-38632-6 (1985)

  22. 22.

    Jiang, T.: Maxima of entries of Haar distributed matrices. Probab. Theory Related Fields 131(1), 121–144 (2005)

  23. 23.

    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)

  24. 24.

    Knowles, A., Yin, J.: The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure Appl. Math. 66(11), 1663–1750 (2013)

  25. 25.

    Knowles, A., Yin, J.: The outliers of a deformed Wigner matrix. Ann. Probab. 42(5), 1980–2031 (2014)

  26. 26.

    O’Rourke, S., Renfrew, D.: Low rank perturbations of large elliptic random matrices. Electron. J. Probab. 19(43), 65 (2014)

  27. 27.

    Péché, S.: The largest eigenvalue of small rank perturbations of Hermitian random matrices. Prob. Theory Relat. Fields 134, 127–173 (2006)

  28. 28.

    Rudelson, M., Vershynin, R.: Invertibility of random matrices: unitary and orthogonal perturbations. J. Amer. Math. Soc. 27, 293–338 (2014)

  29. 29.

    Tao, T.: Topics in random matrix theory, Graduate Studies in Mathematics, AMS (2012)

  30. 30.

    Tao, T.: Outliers in the spectrum of i.i.d. matrices with bounded rank perturbations. Probab. Theory Related Fields 155(1–2), 231–263 (2013)

  31. 31.

    Zhang, C., Qiu, R.C.: Data Modeling with Large Random Matrices in a Cognitive Radio Network Testbed: initial experimental demonstrations with 70 Nodes. arXiv:1404.3788

Download references

Acknowledgments

We would like to thank J. Novak for discussions on Weingarten calculus.

Author information

Correspondence to Florent Benaych-Georges.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Benaych-Georges, F., Rochet, J. Outliers in the Single Ring Theorem. Probab. Theory Relat. Fields 165, 313–363 (2016). https://doi.org/10.1007/s00440-015-0632-x

Download citation

Keywords

  • Random matrices
  • Spiked models
  • Extreme eigenvalue statistics
  • Gaussian fluctuations
  • Ginibre matrices

Mathematics Subject Classification

  • 15A52
  • 60F05