An extension of the Ising-Curie-Weiss model of self-organized criticality with long range interactions

TL;DR

This paper extends a self-organized criticality model based on the Ising-Curie-Weiss framework by exploring how varying the interaction range affects the system's behavior, revealing mean-field and short-range interaction regimes.

Contribution

It introduces a variable-range interaction into the model, analyzing the transition from mean-field to short-range behavior in self-organized criticality.

Findings

Long-range interactions (order n) replicate mean-field behavior.

Short-range interactions (nearest neighbor) alter the critical behavior.

The model's behavior depends critically on the interaction range d(n).

Abstract

In [CG16], 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. In this article, we build upon this model by replacing the mean-field interaction of [CG16] with a one-dimensional interaction with a certain range d(n) which varies as a function of the number n of particles. In the Gaussian case, we show that, for a very long range of interaction (d(n) of order n), the model exhibits the same behaviour as in the mean-field case, whereas in the case of a nearest neighbour interaction (d(n) = 1), the behaviour highlighted by Cerf and Gorny breaks out.