Eigenvector dynamics 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 ideas of [O. Ledoit and S. Peche, Eigenvectors of some large sample covariance matrix ensembles, Probab. Theory Related Fields151 (2011) 233]. Finally, we give a robust derivation of a result obtained in [R. Allez and J.-P. Bouchaud, Eigenvector dynamics: General theory and some applications, Phys. Rev. E86 (2012) 046202] to study eigenspace dynamics in a semi-perturbative regime.

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