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Local Central Limit Theorem for Multi-group Curie–Weiss Models

Author

Listed:
  • Michael Fleermann

    (FernUniversität in Hagen)

  • Werner Kirsch

    (FernUniversität in Hagen)

  • Gabor Toth

    (FernUniversität in Hagen)

Abstract

We study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is, for the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high-temperature regime.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01122-4
    DOI: 10.1007/s10959-021-01122-4
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    References listed on IDEAS

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    1. 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.
    2. William A. Brock & Steven N. Durlauf, 2001. "Discrete Choice with Social Interactions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 68(2), pages 235-260.
    3. Löwe, Matthias & Schubert, Kristina & Vermet, Franck, 2020. "Multi-group binary choice with social interaction and a random communication structure—A random graph approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).
    4. Pierluigi Contucci & Stefano Ghirlanda, 2007. "Modeling society with statistical mechanics: an application to cultural contact and immigration," Quality & Quantity: International Journal of Methodology, Springer, vol. 41(4), pages 569-578, August.
    5. 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.
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