The Annals of Probability

A Curie–Weiss model of self-organized criticality

Raphaël Cerf and Matthias Gorny

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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 $n^{3/4}$, and the limiting law is $C\exp(-\lambda x^{4})\,dx$ where $C$ and $\lambda$ are suitable positive constants.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 444-478.

Received: June 2013
Revised: January 2014
First available in Project Euclid: 2 February 2016

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Ising Curie–Weiss self-organized criticality Laplace’s method


Cerf, Raphaël; Gorny, Matthias. A Curie–Weiss model of self-organized criticality. Ann. Probab. 44 (2016), no. 1, 444--478. doi:10.1214/14-AOP978.

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