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 $A$?