An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range

TL;DR

This paper extends a self-organized criticality model from mean-field to finite-range interactions, revealing a sharp threshold at interaction range proportional to n^{3/4} that affects fluctuation behavior.

Contribution

It introduces a finite-range interaction into the self-organized criticality model and identifies the critical interaction range threshold affecting fluctuations.

Findings

Self-critical behavior persists for interaction ranges much larger than n^{3/4}.

Different fluctuation regimes occur when the interaction range is below or above n^{3/4}.

The threshold at n^{3/4} is sharp, with distinct fluctuation laws.

Abstract

In arXiv:1301.6911, Cerf and Gorny constructed a model of self-organized criticality, by introducing an automatic control of the temperature parameter in the generalized Ising Curie-Weiss model. The fluctuations of the magnetization of this spin model are of order $n^{3/4}$ with a limiting law of the form $C\exp(-x^4)$, as in the critical regime of the Curie-Weiss model. In this article, we build upon this model by replacing the mean-field interaction with a one-dimensional interaction with a certain range $r_n$ which varies as a function of the number $n$ of particles. In the Gaussian case, we show that the self-critical behaviour observed in the mean-field case extends to interaction ranges $r_n\gg n^{3/4}$ and we show that this threshold is sharp, with different fluctuations when the interaction range is of order of $n^{3/4}$ or smaller than $n^{3/4}$.