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The almost sure semicircle law for random band matrices with dependent entries

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  • Fleermann, Michael
  • Kirsch, Werner
  • Kriecherbauer, Thomas

Abstract

We analyze the empirical spectral distribution of random periodic band matrices with correlated entries. The correlation structure we study was first introduced in Hochstättler et al. (2015) by Hochstättler, Kirsch and Warzel, who named their setup almost uncorrelated and showed convergence to the semicircle distribution in probability. We strengthen their results which turn out to be also valid almost surely. Moreover, we extend them to band matrices. Sufficient conditions for convergence to the semicircle law both in probability and almost surely are provided. In contrast to convergence in probability, almost sure convergence seems to require a minimal growth rate for the bandwidth in the correlated case. Examples that fit our general setup include Curie–Weiss distributed, correlated Gaussian, and as a special case, independent entries.

Suggested Citation

  • Fleermann, Michael & Kirsch, Werner & Kriecherbauer, Thomas, 2021. "The almost sure semicircle law for random band matrices with dependent entries," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 172-200.
    • Handle: RePEc:eee:spapps:v:131:y:2021:i:c:p:172-200
    DOI: 10.1016/j.spa.2020.09.004
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    References listed on IDEAS

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    1. Banna, Marwa & Merlevède, Florence & Peligrad, Magda, 2015. "On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2700-2726.
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    Cited by:

    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. 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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