Suppose $D$ is a diagonal matrix of size $n \times n$ with diagonal elements $D_{ii}$ which are independent standard centered Gaussian random variables. Then consider a matrix $J$ such that its elements $J_{ij}$ are independent centered Gaussian variables with variance $\sigma^2/n$.

The question is: what is the limiting eigenvalue distribution of $A=D+J$?

In particular, has this distribution a bounded support ?