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Semicircle Law for a Matrix Ensemble with Dependent Entries

Author

Listed:
  • Winfried Hochstättler

    (FernUniversität in Hagen)

  • Werner Kirsch

    (FernUniversität in Hagen)

  • Simone Warzel

    (Technische Universität München)

Abstract

We study ensembles of random symmetric matrices whose entries exhibit certain correlations. Examples are distributions of Curie–Weiss type. We provide a criterion on the correlations ensuring the validity of Wigner’s semicircle law for the eigenvalue distribution measure. In case of Curie–Weiss distributions, this criterion applies above the critical temperature (i.e., $$\beta \,

Suggested Citation

  • Winfried Hochstättler & Werner Kirsch & Simone Warzel, 2016. "Semicircle Law for a Matrix Ensemble with Dependent Entries," Journal of Theoretical Probability, Springer, vol. 29(3), pages 1047-1068, September.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:3:d:10.1007_s10959-015-0602-3
    DOI: 10.1007/s10959-015-0602-3
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    References listed on IDEAS

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    1. Olga Friesen & Matthias Löwe, 2013. "The Semicircle Law for Matrices with Independent Diagonals," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1084-1096, December.
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    Cited by:

    1. Werner Kirsch & Thomas Kriecherbauer, 2018. "Semicircle Law for Generalized Curie–Weiss Matrix Ensembles at Subcritical Temperature," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2446-2458, December.
    2. Sanders, Jaron & Van Werde, Alexander, 2023. "Singular value distribution of dense random matrices with block Markovian dependence," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 453-504.
    3. Sanders, Jaron & Senen–Cerda, Albert, 2023. "Spectral norm bounds for block Markov chain random matrices," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 134-169.
    4. Michael Fleermann & Werner Kirsch & Gabor Toth, 2022. "Local Central Limit Theorem for Multi-group Curie–Weiss Models," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2009-2019, September.

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    1. Sanders, Jaron & Van Werde, Alexander, 2023. "Singular value distribution of dense random matrices with block Markovian dependence," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 453-504.
    2. Werner Kirsch & Thomas Kriecherbauer, 2018. "Semicircle Law for Generalized Curie–Weiss Matrix Ensembles at Subcritical Temperature," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2446-2458, December.

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