Central limit theorems for eigenvalues of deformations of Wigner matrices

Abstract

In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and sufficient condition on the associated normalized eigenvector so that the fluctuations of the corresponding eigenvalue of the deformed model are universal. RésuméDans ce papier, nousétudions les fluctuations des valeurs propres extrémales d'une matrice de Wigner hermitienne (resp. symétrique) déformée par une perturbation de rang fini dont les valeurs propres non nulles sont fixées, dans le cas où ces valeurs propres extrémales se détachent du reste du spectre. Nous décrivons des situations générales d'universalité ou de non-universalité des fluctuations correspondant au caractère localisé ou délocalisé des vecteurs propres de la perturbation. Lorsque l'une des valeurs propres de la perturbation est de multiplicité un, nousétablissons de plus une condition nécessaire et suffisante sur le vecteur propre associé pour que les fluctuations de la valeur propre correspondante du modèle déformé soient universelles.Mathematics Subject Classification (2010): 60B20, 15A18, 60F05 .

The Isotropic Semicircle Law and Deformation of Wigner Matrices

“…Our main results are deviation estimates on the eigenvalue locations (Theorem 2.7) and the distribution of the outliers (Theorem 2.14). In subsequent remarks we discuss some special cases of interest, in particular making the link to the previous results of [11,12,13,32].…”

“…Such models were first investigated by Füredi and Komlós [29]. Subsequently, much progress [5,6,7,11,12,13,28,32] has been made in the analysis of the spectrum of such deformed matrix models. See, e.g., [32] for a review of recent developments.…”

Correction to: Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model with Ferromagnetic Interaction

“…See also Theorem 1.5 of [3] and Theorem 3.4 of [1]. Together with Theorem 1.5 (iii) of the original article, (3) implies that 1 …”

Outliers in the Single Ring Theorem

“…Thus A n + P n has two outliers λ n := θ p 1,1 λ 1,1 and λ n := θ p 2,1 λ 1,1 and one can compute the numbers K , σ, σ of (13), (14) and get Table 1 Comparison between theoretical asymptotic formulas (16) and (17) and a Monte-Carlo numerical computation made out of 10 3 matrices with size n = 10 3 …”

“…This phenomenon has already been well understood in the Hermitian case. It was shown under several hypotheses in [7][8][9][10][13][14][15]17,24,25,27] that for a large random Hermitian matrix, if the strength of the added perturbation is above a certain threshold, then the extreme eigenvalues of the perturbed matrix deviate at a macroscopic distance from the bulk (such eigenvalues are usually called outliers) and have well understood fluctuations, otherwise they stick to the bulk and fluctuate as those of the non-perturbated matrix (this phenomenon is called the BBP phase transition, named after the authors of [3], who first brought it to light for empirical covariance matrices). Also, Tao, O'Rourke, Renfrew, Bordenave and Capitaine studied a non-Hermitian case: in [11,26,30] they considered spiked i.i.d.…”