The Isotropic Semicircle Law and Deformation of Wigner Matrices

Understand

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 $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$.