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The Cramér condition for the Curie-Weiss model of SOC. (English) Zbl 1366.60108

Summary: We pursue the study of the Curie-Weiss model of self-organized criticality we designed in [R. Cerf and the author, Ann. Probab. 44, No. 1, 444–478 (2016; Zbl 1342.60161)]. We extend our results to more general interaction functions and we prove that, for a class of symmetric distributions satisfying a Cramér condition (C) and some integrability hypothesis, the sum \(S_{n}\) of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order \(n^{3/4}\) and the limiting law is \(k\exp(-\lambda x^{4})dx\) where \(k\) and \(\lambda\) are suitable positive constants. In [loc. cit.], we obtained these results only for distributions having an even density.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
82B27 Critical phenomena in equilibrium statistical mechanics

Citations:

Zbl 1342.60161
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References:

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