Outliers in the Single Ring Theorem

50 pagesThis 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 by 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 asymptotically Gaussian, isotropic on the complex plane with a given variance. 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