IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v32y2019i3d10.1007_s10959-017-0790-0.html
   My bibliography  Save this article

Empirical Spectral Distribution of a Matrix Under Perturbation

Author

Listed:
  • Florent Benaych-Georges

    (Université Paris Descartes)

  • Nathanaël Enriquez

    (Université Paris-Sud)

  • Alkéos Michaïl

    (Université Paris Descartes)

Abstract

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.

Suggested Citation

  • Florent Benaych-Georges & Nathanaël Enriquez & Alkéos Michaïl, 2019. "Empirical Spectral Distribution of a Matrix Under Perturbation," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1220-1251, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-017-0790-0
    DOI: 10.1007/s10959-017-0790-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-017-0790-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-017-0790-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Romain Allez & Jean-Philippe Bouchaud, 2012. "Eigenvector dynamics: general theory and some applications," Papers 1203.6228, arXiv.org, revised Jul 2012.
    2. Joel Bun & Romain Allez & Jean-Philippe Bouchaud & Marc Potters, 2015. "Rotational invariant estimator for general noisy matrices," Papers 1502.06736, arXiv.org, revised Oct 2016.
    3. repec:dau:papers:123456789/10916 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sumanjay Dutta & Shashi Jain, 2023. "Precision versus Shrinkage: A Comparative Analysis of Covariance Estimation Methods for Portfolio Allocation," Papers 2305.11298, arXiv.org.
    2. Ding, Xiucai & Ji, Hong Chang, 2023. "Spiked multiplicative random matrices and principal components," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 25-60.
    3. Sebastien Valeyre, 2020. "Refined model of the covariance/correlation matrix between securities," Papers 2001.08911, arXiv.org.
    4. Jean-Philippe Bouchaud & Iacopo Mastromatteo & Marc Potters & Konstantin Tikhonov, 2022. "Excess Out-of-Sample Risk and Fleeting Modes," Papers 2205.01012, arXiv.org.
    5. Abdullah Karasan & Ozge Sezgin Alp, 2025. "Signal from Noise Signal from Noise: A Neural Network-Based Denoising Approach for Measuring Global Financial Spillovers," Papers 2509.01156, arXiv.org.
    6. Sebastien Valeyre & Denis Grebenkov & Sofiane Aboura & Francois Bonnin, 2016. "Should employers pay their employees better? An asset pricing approach," Papers 1602.00931, arXiv.org, revised Oct 2016.
    7. Lamia Lamrani & Christian Bongiorno & Marc Potters, 2025. "Optimal Data Splitting for Holdout Cross-Validation in Large Covariance Matrix Estimation," Papers 2503.15186, arXiv.org, revised Sep 2025.
    8. Gautier Marti & Frank Nielsen & Miko{l}aj Bi'nkowski & Philippe Donnat, 2017. "A review of two decades of correlations, hierarchies, networks and clustering in financial markets," Papers 1703.00485, arXiv.org, revised Nov 2020.
    9. Heckens, Anton J. & Guhr, Thomas, 2022. "New collectivity measures for financial covariances and correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    10. Ajay Singh & Dinghai Xu, 2016. "Random matrix application to correlations amongst the volatility of assets," Quantitative Finance, Taylor & Francis Journals, vol. 16(1), pages 69-83, January.
    11. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe De Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Documents de travail du Centre d'Economie de la Sorbonne 19022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    12. Bongiorno, Christian & Challet, Damien, 2023. "Non-linear shrinkage of the price return covariance matrix is far from optimal for portfolio optimization," Finance Research Letters, Elsevier, vol. 52(C).
    13. Christian Bongiorno & Damien Challet, 2021. "Covariance matrix filtering with bootstrapped hierarchies," PLOS ONE, Public Library of Science, vol. 16(1), pages 1-13, January.
    14. Charles-Albert Lehalle & Guillaume Simon, 2021. "Portfolio selection with active strategies: how long only constraints shape convictions," Journal of Asset Management, Palgrave Macmillan, vol. 22(6), pages 443-463, October.
    15. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2020. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Post-Print hal-02508748, HAL.
    16. Yan Zhang & Jiyuan Tao & Zhixiang Yin & Guoqiang Wang, 2022. "Improved Large Covariance Matrix Estimation Based on Efficient Convex Combination and Its Application in Portfolio Optimization," Mathematics, MDPI, vol. 10(22), pages 1-15, November.
    17. Rama Cont & Lakshithe Wagalath, 2014. "Institutional Investors and the Dependence Structure of Asset Returns," Working Papers 2014-ACF-01, IESEG School of Management.
    18. Christian Bongiorno & Damien Challet, 2023. "The Oracle estimator is suboptimal for global minimum variance portfolio optimisation," Post-Print hal-03491913, HAL.
    19. Lin Xiao & Arash Sioofy Khoojine, 2024. "Dynamic Anomaly Detection in the Chinese Energy Market During Financial Turbulence Using Ratio Mutual Information and Crude Oil Price Movements," Energies, MDPI, vol. 17(23), pages 1-22, November.
    20. Lamia Lamrani & Beno^it Collins & Jean-Philippe Bouchaud, 2025. "Holdout cross-validation for large non-Gaussian covariance matrix estimation using Weingarten calculus," Papers 2509.13923, arXiv.org.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-017-0790-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.