We consider a square random matrix of the form A + Y, where A is deterministic and Y is invariant under the left and right actions of the unitary group. Then single ring theorem (due to Guionnet, Krishnapur and Zeitouni) says that the eigenvalue distribution of Y converges to the Brown measure of a R-diagonal operator T. Under certain conditions, we show that the eigenvalue distribution of A+Y converges weakly to the Brown measure of the operator a+T, where represents the limit of A. If A has some eigenvalues outside of the support of the limit probability measure, we show that A+Y may or may not produce outliers in some neighborhood of these eigenvalues of A, depending on their locations. This result extends a work of Benaych-Georges and Rochet for the case when A has a finite rank. Joint works with Hari Bercovici, Ching-Wei Ho, and Zhi Yin.
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