article
Extremal eigenvalues and eigenvectors of deformed Wigner matrices
published
yes
Jioon
Lee
author
Kevin
Schnelli
author 434AD0AE-F248-11E8-B48F-1D18A9856A870000-0003-0954-3231
LaEr
department
Random matrices, universality and disordered quantum systems
project
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.
Springer2016
eng
Probability Theory and Related Fields10.1007/s00440-014-0610-8
1641-2165 - 241
J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.
J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” <i>Probability Theory and Related Fields</i>, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.
Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” <i>Probability Theory and Related Fields</i>, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:<a href="https://doi.org/10.1007/s00440-014-0610-8">10.1007/s00440-014-0610-8</a>.
Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.
Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. <i>Probability Theory and Related Fields</i>. Springer. <a href="https://doi.org/10.1007/s00440-014-0610-8">https://doi.org/10.1007/s00440-014-0610-8</a>
Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” <i>Probability Theory and Related Fields</i>. Springer, 2016. <a href="https://doi.org/10.1007/s00440-014-0610-8">https://doi.org/10.1007/s00440-014-0610-8</a>.
Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. <i>Probability Theory and Related Fields</i>. 2016;164(1-2):165-241. doi:<a href="https://doi.org/10.1007/s00440-014-0610-8">10.1007/s00440-014-0610-8</a>
18812018-12-11T11:54:31Z2024-10-09T20:56:36Z