##
**The outliers of a deformed Wigner matrix.**
*(English)*
Zbl 1306.15034

Let \(H\) be a Wigner matrix, i.e., an \(N\times N\) matrix whose entries are independently up to symmetry constraints. Let us consider an additive deformation of \(H\) by a finite-rank matrix \(A\) belonging to the same symmetric class as \(H\). It is very well known (Weyl’s theorem) that such a deformation does not influence the global statistics of the eigenvalues when \(N\rightarrow\infty\). Indeed, the eigenvalue densities of the deformed matrix \(H+A\) and the initial matrix \(H\) have the same large-scale asymptotics and they satisfy the semicircle law. Nevertheless, the individual eigenvalues can change under such a deformation. More precisely, such deformed matrices can exhibit outliers, i.e., eigenvalues detached from the bulk spectrum. For rank-one perturbations, they have been first analyzed by Z. Füredi and J. Komloś [Combinatorica 1, 233–241 (1981; Zbl 0494.15010)]. The creation of an outlier is associated with a sharp transition where the magnitude of an eigenvalue \(d_{k}\) of \(A\) exceeds the threshold 1 (assuming the spectrum of \(H\) is asymptotically given by the interval \([-2,2]\). As \(d_{k}\) becomes larger than \(1\), the largest nonoutlier eigenvalue of \(H+A\) detaches form the bulk spectrum and becomes an outlier. This transition is conjectured to take place on the scale \(d_{k} -1 \sim N^{-1/3}\). This scale was established for the special cases where \(H\) is Gaussian (the Gaussian orthogonal ensemble and the Gaussian unitary ensemble, respectively).

A. Knowles and J. Yin [Commun. Pure Appl. Math. 66, No. 11, 1663–1749 (2013; Zbl 1290.60004)] considered finite-rank deformations of a Wigner matrix whose entries have subexponential decay. They proved that the nonoutliers of the matrix \(H+A\) stick to the extremal eigenvalues of \(H\) with high precision, provided that each eigenvalue \(d_{k}\) of \(A\) satisfies \(||d_{k}|-1| \geq (\log N)^{C \log \log N} N^{-1/3}\). On the other hand, they analyzed the asymptotic distribution of a single outlier assuming that it is separated from the bulk spectrum \([-2,2]\) by at least \((\log N)^{C \log \log N} N^{-2/3}\) and it does not overlap with any other outlier of \(H+A\).

In the paper under review, a complete description of the joint asymptotic distribution of the outliers is given, assuming that the matrix \(A\) is of fixed rank and its norm is bounded. Thus, overlapping outliers are allowed and the joint asymptotic distribution of all outliers is derived. The distribution of overlapping outliers is more complicated than of the nonoverlapping ones, as overlapping outliers have a level repulsion similar to that among the bulk eigenvalues of Wigner matrices. The mechanism underlying the repulsion among outliers is the same as the one for the eigenvalues of Gaussian unitary ensembles. The Jacobian relating the eigenvalue-eigenvector entries to the matrix entries has a Vandermonde determinant structure and vanishes if two eigenvalues coincide. Due to this level repulsion, overlapping outliers are not asymptotically independent, a fact that is emphasized for nonoverlapping outliers as one of the main and novel results of this contributions. This lack of independence does not arise from the level repulsion but from an interplay between the distribution of \(H\) and the geometry of eigenvectors of \(A\). The main result of the contribution under review shows, under suitable conditions on \(H\) and \(A\), that two outliers can be strongly correlated in the limit \(N\rightarrow\infty\), even if they are far from each other.

The proof relies on the isotropic local semicircle law, that constitutes a generalization of the local semicircle law, in that it gives optimal high-probability estimates on \(\langle \mathbf{v}, (G(z)- m(z))\mathbf{w} \rangle\), where \(\mathbf{v}\) and \(\mathbf{w}\) are arbitrary deterministic vectors, \(m(z)\) denotes the Stieltjes transform of the Wigner semicircle law and \(G(z)\) is the resolvent of \(H\).

A four-step of strategy constitutes one of the main tools. First, the reduction to the distribution of the resolvent \(G\), next the cases of Gaussian and almost Gaussian \(H,\) and, finally, the case of general \(H\). On the other hand, a two-level partitioning of the outliers combined with near-degenerate perturbation theory for eigenvalues is used. The outliers are partitioned into blocks depending on wether they overlap.

A. Knowles and J. Yin [Commun. Pure Appl. Math. 66, No. 11, 1663–1749 (2013; Zbl 1290.60004)] considered finite-rank deformations of a Wigner matrix whose entries have subexponential decay. They proved that the nonoutliers of the matrix \(H+A\) stick to the extremal eigenvalues of \(H\) with high precision, provided that each eigenvalue \(d_{k}\) of \(A\) satisfies \(||d_{k}|-1| \geq (\log N)^{C \log \log N} N^{-1/3}\). On the other hand, they analyzed the asymptotic distribution of a single outlier assuming that it is separated from the bulk spectrum \([-2,2]\) by at least \((\log N)^{C \log \log N} N^{-2/3}\) and it does not overlap with any other outlier of \(H+A\).

In the paper under review, a complete description of the joint asymptotic distribution of the outliers is given, assuming that the matrix \(A\) is of fixed rank and its norm is bounded. Thus, overlapping outliers are allowed and the joint asymptotic distribution of all outliers is derived. The distribution of overlapping outliers is more complicated than of the nonoverlapping ones, as overlapping outliers have a level repulsion similar to that among the bulk eigenvalues of Wigner matrices. The mechanism underlying the repulsion among outliers is the same as the one for the eigenvalues of Gaussian unitary ensembles. The Jacobian relating the eigenvalue-eigenvector entries to the matrix entries has a Vandermonde determinant structure and vanishes if two eigenvalues coincide. Due to this level repulsion, overlapping outliers are not asymptotically independent, a fact that is emphasized for nonoverlapping outliers as one of the main and novel results of this contributions. This lack of independence does not arise from the level repulsion but from an interplay between the distribution of \(H\) and the geometry of eigenvectors of \(A\). The main result of the contribution under review shows, under suitable conditions on \(H\) and \(A\), that two outliers can be strongly correlated in the limit \(N\rightarrow\infty\), even if they are far from each other.

The proof relies on the isotropic local semicircle law, that constitutes a generalization of the local semicircle law, in that it gives optimal high-probability estimates on \(\langle \mathbf{v}, (G(z)- m(z))\mathbf{w} \rangle\), where \(\mathbf{v}\) and \(\mathbf{w}\) are arbitrary deterministic vectors, \(m(z)\) denotes the Stieltjes transform of the Wigner semicircle law and \(G(z)\) is the resolvent of \(H\).

A four-step of strategy constitutes one of the main tools. First, the reduction to the distribution of the resolvent \(G\), next the cases of Gaussian and almost Gaussian \(H,\) and, finally, the case of general \(H\). On the other hand, a two-level partitioning of the outliers combined with near-degenerate perturbation theory for eigenvalues is used. The outliers are partitioned into blocks depending on wether they overlap.

Reviewer: Francisco Marcellán (Leganes)

### Keywords:

random matrices; universality; deformation; outliers; Wigner matrix; eigenvalue densities; asymptotics; semicircle law; bulk spectrum; Gaussian orthogonal ensemble; Gaussian unitary ensemble; eigenvector; Stieltjes transform
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\textit{A. Knowles} and \textit{J. Yin}, Ann. Probab. 42, No. 5, 1980--2031 (2014; Zbl 1306.15034)

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