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Delocalization at Small Energy for Heavy-Tailed Random Matrices

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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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Author information

Authors and Affiliations

  1. CNRS and Université Toulouse III, Toulouse, France

    Charles Bordenave

  2. CNRS and École Normale Supérieure de Lyon, Lyon, France

    Alice Guionnet

  3. MIT, Cambridge, MA, USA

    Alice Guionnet

Authors
  1. Charles Bordenave
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  2. Alice Guionnet
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Correspondence to Charles Bordenave.

Additional information

Communicated by L. Erdös

Charles Bordenave: Partially supported by Grants ANR-14-CE25-0014 and ANR-16-CE40-0024-01. Alice Guionnet: Partially supported by the Simons Foundation and by NSF Grant DMS-1307704.

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Bordenave, C., Guionnet, A. Delocalization at Small Energy for Heavy-Tailed Random Matrices. Commun. Math. Phys. 354, 115–159 (2017). https://doi.org/10.1007/s00220-017-2914-x

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