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Outliers in the spectrum of large deformed unitarily invariant models

We investigate the asymptotic behavior of the eigenvalues of the sum A+U*BU, where A and B are deterministic N by N Hermitian matrices having respective limiting compactly supported distributions μ, ν, and U is a random N by N unitary matrix distributed according to Haar measure. We assume that A has a fixed number of fixed eigenvalues (spikes) outside the support of μ, whereas the distances between the other eigenvalues of A and the support of μ, and between the eigenvalues of B and the support of ν, uniformly go to zero as N goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of A+U*BU outside the support of the limiting spectral measure, called outliers. This phenomenon is fully described in terms of free probability involving the subordination function related to the free additive convolution of μ and ν. Only finite rank perturbations had been considered up to now.