Eigenvectors dynamic and local density of states under free addition

We investigate the evolution of a given eigenvector of a symmetric (deterministic or random) matrix under the addition of a matrix in the Gaussian orthogonal ensemble. We quantify the overlap between this single vector with the eigenvectors of the initial matrix and identify precisely a "Cauchy-flight" regime. In particular, we compute the local density of this vector in the eigenvalues space of the initial matrix. Our results are obtained in a non perturbative setting and are derived using the idea of Ledoit and Péché in [11]. Finally, we revisit our former results on the eigenspace dynamics giving a robust derivation of a result obtained in [1] in a semi-perturbative regime.