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Two Groups in a Curie–Weiss Model with Heterogeneous Coupling

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  • Werner Kirsch

    (FernUniversität Hagen)

  • Gabor Toth

    (FernUniversität Hagen)

Abstract

We discuss a Curie–Weiss model with two groups with different coupling constants within and between groups. For the total magnetisations in each group, we show bivariate laws of large numbers and a central limit theorem which is valid in the high-temperature regime. In the critical regime, the total magnetisation normalised by $$N^{3/4}$$ N 3 / 4 converges to a non-trivial distribution which is not Gaussian, just as in the single-group Curie–Weiss model. Finally, we prove a kind of a ‘law of large numbers’ in the low-temperature regime, more precisely we prove that the empirical magnetisation converges in distribution to a mixture of two Dirac measures.

Suggested Citation

  • Werner Kirsch & Gabor Toth, 2020. "Two Groups in a Curie–Weiss Model with Heterogeneous Coupling," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2001-2026, December.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00933-w
    DOI: 10.1007/s10959-019-00933-w
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    References listed on IDEAS

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    1. 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.
    2. Kleiber, Christian & Stoyanov, Jordan, 2013. "Multivariate distributions and the moment problem," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 7-18.
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