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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
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    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
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