A Curie-Weiss model of self-organized criticality

We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. For a class of symmetric distributions whose density satisfies some integrability conditions, we prove that the sum $S_n$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order $n^{3\mathrm{/}4}$, and the limiting law is $C\exp (-\lambda x^4)dx$ where $C$ and $\lambda$ are suitable positive constants.