  We consider a full rank deformation of the GUE $W_N+A_N$ where $A_N$ is a
full rank Hermitian matrix of size $N$ and $W_N$ is a GUE. The empirical
eigenvalue distribution $\mu_{A_N}$ of $A_N$ converges to a probability
distribution $\nu$. We identify all the possible limiting eigenvalue statistics
at the edges of the spectrum, including outliers, edges and merging points of
connected components of the limiting spectrum. The results are stated in terms
of a deterministic equivalent of the empirical eigenvalue distribution of
$W_N+A_N$, namely the free convolution of the semi-circle distribution and the
empirical eigenvalues distribution of $A_N$.
