---
_id: '1881'
abstract:
- lang: eng
text: '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. '
acknowledgement: "Most of the presented work was obtained while Kevin Schnelli was
staying at the IAS with the support of\r\nThe Fund For Math."
author:
- first_name: Jioon
full_name: Lee, Jioon
last_name: Lee
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
orcid: 0000-0003-0954-3231
citation:
ama: 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/s00440-014-0610-8
apa: Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of
deformed Wigner matrices. *Probability Theory and Related Fields*. Springer.
https://doi.org/10.1007/s00440-014-0610-8
chicago: Lee, 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.
ieee: 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.
ista: Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed
Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.
mla: 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.
short: J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.
corr_author: '1'
date_created: 2018-12-11T11:54:31Z
date_published: 2016-02-01T00:00:00Z
date_updated: 2024-10-09T20:56:36Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s00440-014-0610-8
ec_funded: 1
intvolume: ' 164'
issue: 1-2
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1310.7057
month: '02'
oa: 1
oa_version: Preprint
page: 165 - 241
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Probability Theory and Related Fields
publication_status: published
publisher: Springer
publist_id: '5215'
quality_controlled: '1'
scopus_import: 1
status: public
title: Extremal eigenvalues and eigenvectors of deformed Wigner matrices
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 164
year: '2016'
...