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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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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DOI: https://doi.org/10.1007/s00220-017-2914-x