  We consider a square random matrix of size N of the form A + Y where A is
deterministic and Y has iid entries with variance 1/N. Under mild assumptions,
as N grows, the empirical distribution of the eigenvalues of A+Y converges
weakly to a limit probability measure \beta on the complex plane. This work is
devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the
complement of the support of \beta. Even in the simplest cases, a variety of
interesting phenomena can occur. As in earlier works, we give a sufficient
condition to guarantee that outliers are stable and provide examples where
their fluctuations vary with the particular distribution of the entries of Y or
the Jordan decomposition of A. We also exhibit concrete examples where the
outlier eigenvalues converge in distribution to the zeros of a Gaussian
analytic function.
