2010/09/01 by Florent Benaych-Georges, Benaych-Georges, Florent, Alice Guionnet +3 · 5 citations
Mathematics · #Random Matrices and Applications #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics
paper · doi:10.48550/arxiv.1009.0145
Consider a deterministic self-adjoint matrix Xn with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix Xn so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when Xn is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.