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arXiv · 1301.6911

A Curie-Weiss model of self-organized criticality

Raphaël Cerf·Matthias Gorny
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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. 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 $C\exp(-λx^4)\,dx$ where $C$ and $λ$ are suitable positive constants.

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Raphaël Cerf, Matthias Gorny. 2016-02-09. A Curie-Weiss model of self-organized criticality. https://doi.org/10.1214/14-aop978

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