Empirical spectral distribution of a matrix under perturbation

2017/01/10 by Florent Benaych-Georges, Benaych-Georges, Florent, Nathanaël Enriquez +3
Mathematics · #15A52 #46L54 #47A55 #60B20 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:15A52 #msc:46L54 #msc:47A55 #msc:60B20

paper · pdf · doi:10.48550/arxiv.1701.02597

25 pages, 4 figures. In the last version, Remark 1 was added and several points were clarified

arxiv created 2017/10/02 · arxiv updated 2017/10/03

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 one 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 either related to the one-dimensional Gaussian free field or to free probability theory.

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