Delocalization at Small Energy for Heavy-Tailed Random Matrices

Abstract

We prove that the eigenvectors associated to small enough eigenvalues of an heavy-tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity. The delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of L 4 and L 2 -norms of vectors. In contrast, as a consequence of a previous result, for random matrices with sufficiently heavy tails, the eigenvectors associated to large enough eigenvalues are localized according to the same criterion. The proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix.

Eigenvectors of random matrices: A survey

“…Hence, we expect this eigenvector to be concentrated on at most two coordinates. This heuristic has led to a number of results regarding the eigenvalues and eigenvectors of heavy-tailed Wigner matrices; we refer the reader to [3,7,8,12,13,17,68,69] and references therein for further details and additional results.…”

Delocalization Transition for Critical Erdős–Rényi Graphs

“…Heavy-tailed Wigner matrices, or Lévy matrices, whose entries have α-stable laws for 0 < α < 2, were proposed in [24] as a simple model that exhibits a transition in the localization of its eigenvectors; we refer to [3] for a summary of the predictions from [24,53]. In [18,19] it was proved that for energies in a compact interval around the origin, eigenvectors are weakly delocalized, and for 0 < α < 2/3 for energies far enough from the origin, eigenvectors are weakly localized. In [3], full delocalization was proved in a compact interval around the origin, and the authors even established GOE local eigenvalue statistics in the same spectral region.…”

Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models: Wigner–Weisskopf approach

“…Within Random Matrix Models, the question of the existence of a non-ergodic delocalized phase has been actually raised more than twenty years ago by Cizeau and Bouchaud [33] in their pioneering work on Random Lévy Matrices, that has attracted a lot of interest among physicists [34][35][36][37][38][39][40] and mathematicians [41][42][43][44][45][46][47][48]. More recently, the Generalized-Rosenzweig-Porter model has been proposed as the simplest matrix model exhibiting a delocalized nonergodic phase with an explicit multifractal spectrum for eigenvectors in [49].…”