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

Abstract

In [10], Cerf and Gorny constructed a model of self-organized criticality, by introducing an automatic control of the temperature parameter in the generalized Ising Curie-Weiss model. The fluctuations of the magnetization of this spin model are of order $n^{3∕4}$ with a limiting law of the form $Cexp(-x^4)$, as in the critical regime of the Curie-Weiss model.

In this article, we build upon this model by replacing the mean-field interaction with a one-dimensional interaction with a certain range $r_n$ which varies as a function of the number n of particles. In the Gaussian case, we show that the self-critical behaviour observed in the mean-field case extends to interaction ranges $r_n\gg n^{3∕4}$ and we show that this threshold is sharp, with different fluctuations when the interaction range is of order of $n^{3∕4}$ or smaller than $n^{3∕4}$.