Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Path-space moderate deviations for a Curie–Weiss model of self-organized criticality

Francesca Collet, Matthias Gorny, and Richard C. Kraaij

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Abstract

The dynamical Curie–Weiss model of self-organized criticality (SOC) was introduced in (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658–678) and it is derived from the classical generalized Curie–Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie–Weiss model of SOC (Ann. Probab. 44 (2016) 444–478) as unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton–Jacobi equations. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical.

Résumé

Le modèle de Curie–Weiss de criticalité auto-organisée dynamique a été construit dans (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658–678) à partir du modèle de Curie–Weiss généralisé. Il s’agit d’un processus de Markov continu dont l’unique mesure invariante est la loi du modèle de Curie–Weiss de criticalité auto-organisée (Ann. Probab. 44 (2016) 444–478). Dans le cas Gaussien, nous étudions les fluctuations modérées de la magnétisation. Nous obtenons un principe de déviations modérées dans l’espace des chemins en utilisant une approche analytique basée sur la convergence de générateurs non-linéaires et sur l’unicité des solutions de viscosité pour des équations de Hamilton–Jacobi associées. Notre résultat montre que, dans une certaine échelle de temps modérée et sans intervention de paramètres extérieurs, le comportement critique de la magnétisation est critique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 765-781.

Dates
Received: 26 January 2018
Revised: 23 November 2018
Accepted: 20 March 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345618

Digital Object Identifier
doi:10.1214/19-AIHP981

Mathematical Reviews number (MathSciNet)
MR4076764

Subjects
Primary: 60F10: Large deviations 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Moderate deviations Interacting particle systems Mean-field interaction Self-organized criticality Hamilton–Jacobi equation Perturbation theory for Markov processes

Citation

Collet, Francesca; Gorny, Matthias; Kraaij, Richard C. Path-space moderate deviations for a Curie–Weiss model of self-organized criticality. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 765--781. doi:10.1214/19-AIHP981. https://projecteuclid.org/euclid.aihp/1584345618


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