Communications in Mathematical Physics

, Volume 354, Issue 1, pp 115–159 | Cite as

Delocalization at Small Energy for Heavy-Tailed Random Matrices

Article

Abstract

We prove that the eigenvectors associated to small enough eigenvalues of a 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.CNRS and Université Toulouse IIIToulouseFrance
  2. 2.CNRS and École Normale Supérieure de LyonLyonFrance
  3. 3.MITCambridgeUSA

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