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Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

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Abstract

We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.

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Authors and Affiliations

  1. MAP 5, UMR CNRS 8145, Université Paris Descartes, 45 rue des Saints-Pères, 75270, Paris Cedex 6, France

    Florent Benaych-Georges & Camille Male

  2. CNRS and École Normale Supéerieure de Lyon, Unité de mathématiques pures et appliquées, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Alice Guionnet

  3. Mathematics Department, MIT, 77 Massachusetts Av, Cambridge, MA, 02139-4307, USA

    Alice Guionnet

Authors
  1. Florent Benaych-Georges
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  2. Alice Guionnet
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  3. Camille Male
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Correspondence to Camille Male.

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Communicated by L. Erdös

Research supported by ANR-08-BLAN-0311-01, Simons Foundation and Fondation Science mathématique de Paris.

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Benaych-Georges, F., Guionnet, A. & Male, C. Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices. Commun. Math. Phys. 329, 641–686 (2014). https://doi.org/10.1007/s00220-014-1975-3

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  • DOI: https://doi.org/10.1007/s00220-014-1975-3

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