  Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral
measure converging to a compactly supported probability measure. We perturb
this matrix by adding a random finite rank matrix, with delocalized
eigenvectors. We show that the joint law of the extreme eigenvalues of the
perturbed model satisfies a large deviation principle in the scale $n$, with a
good rate function given by a variational formula. We tackle both cases when
the extreme eigenvalues of $X_n$ converge to the edges of the support of the
limiting measure and when we allow some eigenvalues of $X_n$, that we call
outliers, to converge out of the bulk. We can also generalise our results to
the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\ud
X,$ for $V$ growing fast enough at infinity and any perturbation of finite
rank.
