  It is known that if one perturbs a large iid random matrix by a bounded rank
error, then the majority of the eigenvalues will remain distributed according
to the circular law. However, the bounded rank perturbation may also create one
or more outlier eigenvalues. We show that if the perturbation is small, then
the outlier eigenvalues are created next to the outlier eigenvalues of the
bounded rank perturbation; but if the perturbation is large, then many more
outliers can be created, and their law is governed by the zeroes of a random
Laurent series with Gaussian coefficients. On the other hand, these outliers
may be eliminated by enforcing a row sum condition on the final matrix.
