The isotropic semicircle law and deformation of Wigner matrices

Antti Knowles Harvard University Jun Yin University of Wisconsin-Madison

Probability mathscidoc:1608.28013

ommunications on Pure & Applied Mathematics, 66, (11), 1663–1749, 2013
We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we identify the law of the outliers and prove that the non-outliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the \emph{isotropic local semicircle law}, which establishes optimal high-probability bounds on the quantity $< v,[(H - z)^{-1} - m(z) 1] w >$, where $m(z)$ is the Stieltjes transform of Wigner's semicircle law and $v, w$ are arbitrary deterministic vectors.
Random matrix, universality, deformation, outliers
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  title={The isotropic semicircle law and deformation of Wigner matrices},
  author={Antti Knowles, and Jun Yin},
  booktitle={ommunications on Pure & Applied Mathematics},
Antti Knowles, and Jun Yin. The isotropic semicircle law and deformation of Wigner matrices. 2013. Vol. 66. In ommunications on Pure & Applied Mathematics. pp.1663–1749.
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