A Dynamical Curie-Weiss Model of SOC: The Gaussian Case

TL;DR

This paper introduces a Markov process that models self-organized criticality (SOC) with a Gaussian case, demonstrating its fluctuations follow a specific scaling and converge to a critical stochastic differential equation.

Contribution

It rigorously establishes a dynamical model of SOC with precise fluctuation scaling and convergence to a critical SDE in the Gaussian case.

Findings

Fluctuations of the sum evolve on a √n time scale.

Space scale of fluctuations is n^{3/4}.

Limiting process solves a critical stochastic differential equation.

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 in arXiv:1301.6911. 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.