There are Isotropic/Anisotropic local laws for Wigner matrices. I have also seen many works such as The Isotropic Semicircle Law and Deformation of Wigner Matrices on analysis of eigenvalues of low rank deformations of the Wigner matrices. Is there any work on local laws for the resolvent of these low rank deformations? In other words analysis of $<x,R^{-1}y>$ such that $R = \frac{(X+P)^T(X+P)}{n} - zI$ and $P$ is for example $\Sigma_{i=1}^{k} d_i v_iv_i^*$ for a fixed $k$ when $n,p \rightarrow \infty$ and $\frac{n}{p} \rightarrow c$. Also, I am mainly interested in large $d_i$ values (order of $\sqrt{p}$).
