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Two Groups in a Curie–Weiss Model with Heterogeneous Coupling

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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 \(N^{3/4}\) 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.

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Acknowledgements

While finishing this paper, we became aware of the papers [8, 9] which contain the above results as special cases. The methods used by those authors are very different from ours. We are grateful to Francesca Collet for drawing our attention to the papers [8, 9]. See also [2, 3]. We would also like to thank Matthias Löwe and Kristina Schubert [18] as well as an unnamed referee for valuable comments which in our opinion improved this paper considerably.

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  1. Fakultät für Mathematik und Informatik, FernUniversität Hagen, 58084, Hagen, Germany

    Werner Kirsch & Gabor Toth

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  1. Werner Kirsch
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  2. Gabor Toth
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Correspondence to Werner Kirsch.

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Kirsch, W., Toth, G. Two Groups in a Curie–Weiss Model with Heterogeneous Coupling. J Theor Probab 33, 2001–2026 (2020). https://doi.org/10.1007/s10959-019-00933-w

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  • DOI: https://doi.org/10.1007/s10959-019-00933-w

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