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

Nicolas Forien

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 $n^{3/4}$.

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

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 with a limiting law of the form , 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 which varies as a function of the number of particles. In the Gaussian case, we show that the self-critical behaviour observed in the mean-field case extends to interaction ranges and we show that this threshold is sharp, with different fluctuations when the interaction range is of order of or smaller than .

Paper Structure

This paper contains 34 sections, 17 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of the model
  3. Results
  4. Self-critical behaviour in the long range case
  5. Threshold phenomenon
  6. The finite range case
  7. An intermediate regime
  8. Motivation: a Curie-Weiss model of self-organized criticality
  9. Perspectives
  10. More general distributions
  11. The intermediate regime
  12. Different kinds of long range interactions
  13. Random couplings
  14. Sketch of the proof and organization of the paper
  15. Diagonalization of the interaction Hamiltonian
  16. ...and 19 more sections

Key Result

Theorem 1.1

If the interaction range $r_n$ is such that $r_n/n^{3/4}\to\infty$, then under the law $\mu_n$ defined by (eq:defModele), we have the convergence in distribution

Figures (2)

  • Figure 1: In the different regimes studied, if the interaction range is of order $r_n\sim n^a$, then the fluctuations of the magnetization $S_n$ are of order $n^b$, with $b=(1/2+a/3)\wedge(3/4)$.
  • Figure 2: Cauchy's Theorem allows us to replace the integral on the segment $[-M,M]$ by the integral along the three segments $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$.

Theorems & Definitions (28)

  • Theorem 1.1: Long range case
  • Theorem 1.2: Threshold phenomenon
  • Theorem 1.3: Finite range case
  • Theorem 1.4: Intermediate regime
  • Lemma 1.5: Independence of the temperature
  • Corollary 1.6: Behaviour of the temperature
  • Lemma 1.7: Distribution of the self-normalized magnetization
  • Corollary 1.8: Relating the self-normalized magnetization to the behaviour of $A_n$
  • Lemma 1.9: Behaviour of $A_n$ in the long range case
  • Lemma 1.10: Behaviour of $A_n$ at the threshold
  • ...and 18 more