Let $A$ be an $n\times n$ Hermitian matrix, $0\leq \left\|A\right\| \leq 1$, and let $G$ be a Gaussian matrix, i.e. each $G_{i,j}$ is distributed ${\cal N}(0,1)$. What can be said about the distribution of the eigenvalues of $A+ \epsilon \cdot G$, where $\epsilon = 1/poly(n)$.

I would like to know whether there are properties that hold for ANY matrix $A$:

(1) Are the eigenvalues of $A+G$ distributed independently?

(2) Can the variance of the distribution of each eigenvalue of $A+G$ be lower-bounded by some function of $\epsilon$?

If the answer is negative, is there ANY perturbation technique that can yield properties (1) and (2) for any matrix $A$?

Let $A$ be an $n\times n$ Hermitian matrix, with well-separated eigenvalues $\lambda_1 > \lambda_2 ... > \lambda_n$, with $|\lambda_i-\lambda_j|>\epsilon$, for all $i \neq j$. Let $G$ be a Gaussian matrix, i.e. each $G_{i,j}$ is distributed ${\cal N}(0,1)$. What can be said about the distribution of the eigenvalues of $A+ f(\epsilon) \cdot G$, where $f$ is some function $f(\epsilon)<<\epsilon$. Note that the strength of the perturbation is significantly smaller than the inter-eigenvalue distance, so eigenvalue repulsion can be made arbitrarily weak.

I would like to know whether there are properties that hold for ANY such matrix $A$:

(1) Are the eigenvalues of $A+G$ distributed approximately independently?

(2) Can the variance of the distribution of each eigenvalue of $A+G$ be lower-bounded by some function of $\epsilon$?

If the answer is negative, is there ANY perturbation technique that can yield properties (1) and (2) for any such matrix $A$?