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

Abstract

The dynamical Curie-Weiss model of self-organized criticality (SOC) was introduced in [15] 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 [5] 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.Keywords: moderate deviations · interacting particle systems · mean-field interaction · selforganized criticality · Hamilton-Jacobi equation · perturbation theory for Markov processes 2 Model and main result Notation and definitionsBefore starting with the main contents of the paper, we introduce some notation. We start with the definition of good rate-function and of large deviation principle for a sequence of random variables.Definition 2.1. Let (X n ) n∈N * be a sequence of random variables on a Polish space X . Furthermore, consider a function I : X → [0, ∞] and a sequence (r n ) n∈N * of positive numbers such that

Path-space moderate deviations for a class of Curie–Weiss models with dissipation

“…The perturbative argument takes inspiration from the perturbation theory for Markov processes introduced in [26] and it was also used to study path-space moderate deviations for the Curie-Weiss model with random field [8] and with the "self-organized criticality" property [6].…”

“…Remark A.2. Our current set-up is slightly easier than the corresponding set-up in [6,8], in the sense that the sets η −1 n (K q n ) in those papers are non-compact. As an indirect consequence, here we do not have to work with upper and lower limiting operators H † and H ‡ , which greatly simplifies the proof of the moderate deviation principles.…”