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In arXiv:1301.6911, Cerf and Gorny constructed a model of self-organizedcriticality, by introducing an automatic control of the temperature parameterin the generalized Ising Curie-Weiss model. The fluctuations of themagnetization of this spin model are of order $n^{3/4}$ with a limiting law ofthe form $C\exp(-x^4)$, as in the critical regime of the Curie-Weiss model.  In this article, we build upon this model by replacing the mean-fieldinteraction with a one-dimensional interaction with a certain range $r_n$ whichvaries as a function of the number $n$ of particles. In the Gaussian case, weshow that the self-critical behaviour observed in the mean-field case extendsto interaction ranges $r_n\gg n^{3/4}$ and we show that this threshold issharp, with different fluctuations when the interaction range is of order of$n^{3/4}$ or smaller than $n^{3/4}$.

An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range