Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A dynamical Curie–Weiss model of SOC: The Gaussian case

Matthias Gorny

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Abstract

In this paper, we introduce a Markov process whose unique invariant distribution is the Curie–Weiss model of self-organized criticality (SOC) we designed and studied in (Ann. Probab. 44(1):444-478, 2016). In the Gaussian case, we prove rigorously that it is a dynamical model of SOC: the fluctuations of the sum $S_{n}(\cdot)$ of the process evolve in a time scale of order $\sqrt{n}$ and in a space scale of order $n^{3/4}$ and the limiting process is the solution of a “critical” stochastic differential equation.

Résumé

Dans cet article, nous introduisons un processus de Markov dont l’unique distribution invariante est le modèle d’Ising Curie–Weiss de criticalité auto-organisée que nous avons construit et étudié dans (Ann. Probab. 44(1):444-478, 2016). Dans le cas Gaussien, nous montrons rigoureusement qu’il s’agit d’un modèle dynamique de criticalité auto-organisée : les fluctuations de la somme $S_{n}(\cdot)$ du processus évoluent à une vitesse de temps d’ordre $\sqrt{n}$ et à une échelle spatiale d’ordre $n^{3/4}$ et le processus limite est la solution d’une équation différentielle stochastique « critique ».

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 658-678.

Dates
Received: 16 July 2015
Revised: 30 October 2015
Accepted: 10 November 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897740

Digital Object Identifier
doi:10.1214/15-AIHP729

Mathematical Reviews number (MathSciNet)
MR3634269

Zentralblatt MATH identifier
1370.60188

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Ising Curie–Weiss Self-organized criticality Critical fluctuations Langevin diffusion Collapsing processes

Citation

Gorny, Matthias. A dynamical Curie–Weiss model of SOC: The Gaussian case. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 658--678. doi:10.1214/15-AIHP729. https://projecteuclid.org/euclid.aihp/1491897740


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