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Complex Outliers of Hermitian Random Matrices

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  • Jean Rochet

    (MAP5, Université Paris Descartes)

Abstract

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1 ). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.

Suggested Citation

  • Jean Rochet, 2017. "Complex Outliers of Hermitian Random Matrices," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1624-1654, December.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0686-4
    DOI: 10.1007/s10959-016-0686-4
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    References listed on IDEAS

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    1. Benaych-Georges, Florent & Nadakuditi, Raj Rao, 2012. "The singular values and vectors of low rank perturbations of large rectangular random matrices," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 120-135.
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    Cited by:

    1. Patryk Pagacz & Michał Wojtylak, 2022. "Random Perturbations of Matrix Polynomials," Journal of Theoretical Probability, Springer, vol. 35(1), pages 52-88, March.

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