A Curie - Weiss Model of Self-Organized Criticality: The Gaussian Case

#### M. Gorny

2014, v.20, Issue 3, 563-576

ABSTRACT

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. With the help of exact computations, we show that, in the case of a centered Gaussian measure with positive variance $\sigma^2$, the sum $S_n$ of the random variables has fluctuations of order $n^{3/4}$ and that $S_n/n^{3/4}$ converges to the distribution $C \exp(-x^{4}/(4\sigma^4)) dx$ where $C$ is a suitable positive constant.

Keywords: Ising Curie - Weiss,self-organized criticality,Laplace's method