Mean Field Behaviour in a Local Low-Dimensional Model

TL;DR

This paper introduces a mechanism causing mean field behavior in a 2D nonequilibrium model, demonstrating that mean field approximations can be accurate even in low-dimensional, short-range interaction systems.

Contribution

The study reveals a new mechanism for mean field behavior in low-dimensional nonequilibrium systems, supported by a stochastic sandpile model analysis.

Findings

Mean field behavior observed in a 2D non-conservative sandpile model.

Bethe tree approximation closely matches the model's behavior.

Critical behavior resembles mean field percolation despite low dimension.

Abstract

We point out a new mechanism which can lead to mean field type behaviour in nonequilibrium critical phenomena. We demonstrate this mechanism on a two-dimensional model which can be understood as a stochastic and non-conservative version of the abelian sandpile model of Bak {\it et al.} ~\cite{bak}. This model has a second order phase transition whose critical behaviour seems at least partly described by the mean field approximation for percolation, in spite of the low dimension and the fact that all interactions are of short range. Furthermore, the approximation obtained by replacing the lattice by a Bethe tree is very precise in the entire range of the control parameter.