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 Sn of the random variables behaves as in the typical critical generalized Ising Curie.Weiss model. The fluctuations are of order n3/4, and the limiting law is Cexp(..x4)dx where C and . are suitable positive constants.