Large deviations of the extreme eigenvalues of random deformations of matrices

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.