A Curie-Weiss model of self-organized criticality

TL;DR

This paper introduces a modified Curie-Weiss model with automatic temperature control to demonstrate self-organized criticality, providing rigorous mathematical analysis and characterizing the fluctuations and limiting distribution.

Contribution

It presents a new self-organized criticality model based on the Curie-Weiss framework with proven fluctuation behavior and limiting law.

Findings

Fluctuations of the sum are of order n^{3/4}.

Limiting distribution is proportional to exp(-λx^4).

Model exhibits self-organized criticality with rigorous analysis.

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 $C\exp(-\lambda x^4)\,dx$ where $C$ and $\lambda$ are suitable positive constants.