# Extremal eigenvalues and eigenvectors of deformed Wigner matrices

Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.

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*Journal Article*|

*Published*|

*English*

**Scopus indexed**

Author

Lee, Jioon;
Schnelli, Kevin

^{IST Austria}^{}Department

Abstract

We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues.

Publishing Year

Date Published

2016-02-01

Journal Title

Probability Theory and Related Fields

Acknowledgement

Most of the presented work was obtained while Kevin Schnelli was staying at the IAS with the support of
The Fund For Math.

Volume

164

Issue

1-2

Page

165 - 241

IST-REx-ID

### Cite this

Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices.

*Probability Theory and Related Fields*. 2016;164(1-2):165-241. doi:10.1007/s00440-014-0610-8Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices.

*Probability Theory and Related Fields*. Springer. https://doi.org/10.1007/s00440-014-0610-8Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.”

*Probability Theory and Related Fields*. Springer, 2016. https://doi.org/10.1007/s00440-014-0610-8.J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,”

*Probability Theory and Related Fields*, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.”

*Probability Theory and Related Fields*, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:10.1007/s00440-014-0610-8.**All files available under the following license(s):**

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