Eigenvector dynamics under free addition
Abstract
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 "Cauchyflight" 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. Péché, Prob. Th. Rel. Fields, {\bf 151} 233 (2011)]. Finally, we give a robust derivation of a result obtained in [R. Allez and J.P. Bouchaud, Phys. Rev. E {\bf 86}, 046202 (2012)] to study eigenspace dynamics in a semiperturbative regime.
 Publication:

arXiv eprints
 Pub Date:
 January 2013
 arXiv:
 arXiv:1301.4939
 Bibcode:
 2013arXiv1301.4939A
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Statistical Mechanics
 EPrint:
 20 pages, 2 figures