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January 2016 A Curie–Weiss model of self-organized criticality
Raphaël Cerf, Matthias Gorny
Ann. Probab. 44(1): 444-478 (January 2016). DOI: 10.1214/14-AOP978


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.


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Raphaël Cerf. Matthias Gorny. "A Curie–Weiss model of self-organized criticality." Ann. Probab. 44 (1) 444 - 478, January 2016.


Received: 1 June 2013; Revised: 1 January 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1342.60161
MathSciNet: MR3456343
Digital Object Identifier: 10.1214/14-AOP978

Primary: 60F05 , 60K35

Keywords: Ising Curie–Weiss , Laplace’s method , Self-organized criticality

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 1 • January 2016
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