Empirical Spectral Distribution of a Matrix Under Perturbation

Florent Benaych-Georges (), Nathanaël Enriquez () and Alkéos Michaïl ()
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Florent Benaych-Georges: Université Paris Descartes
Nathanaël Enriquez: Université Paris-Sud
Alkéos Michaïl: Université Paris Descartes

Journal of Theoretical Probability, 2019, vol. 32, issue 3, 1220-1251

Abstract: 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.

Keywords: Random matrices; Perturbation theory; Wigner matrices; Band matrices; Hilbert transform; Spectral density; 15A52; 60B20; 47A55; 46L54 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-017-0790-0

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