  This text is about spiked models of non Hermitian random matrices. More
specifically, we consider matrices of the type $A+P$, where the rank of $P$
stays bounded as the dimension goes to infinity and where the matrix $A$ is a
non Hermitian random matrix, satisfying an isotropy hypothesis: its
distribution is invariant under the left and right actions of the unitary
group. The macroscopic eigenvalue distribution of such matrices is governed by
the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We
first prove that if $P$ has some eigenvalues out of the maximal circle of the
single ring, then $A+P$ has some eigenvalues (called outliers) in the
neighborhood of those of $P$, which is not the case for the eigenvalues of $P$
in the inner cycle of the single ring. Then, we study the fluctuations of the
outliers of $A$ around the eigenvalues of $P$ and prove that they are
distributed as the eigenvalues of some finite dimensional random matrices. Such
facts had already been noticed for Hermitian models. More surprising facts are
that outliers can here have very various rates of convergence to their limits
(depending on the Jordan Canonical Form of $P$) and that some correlations can
appear between outliers at a macroscopic distance from each other (a fact
already noticed by Knowles and Yin in the Hermitian case, but only in the case
of non Gaussian models, whereas spiked Gaussian matrices belong to our model
and can have such correlated outliers). Our first result generalizes a previous
result by Tao for matrices with i.i.d. entries, whereas the second one (about
the fluctuations) is new.
