Organized versus self-organized criticality in the abelian sandpile model

TL;DR

This paper investigates the stabilizability of infinite volume configurations in the abelian sandpile model, revealing a phase transition between stabilizable and non-stabilizable states and linking self-organized criticality to this transition.

Contribution

It introduces the concept of stabilizability for infinite configurations and measures, and characterizes the critical behavior of the ASM as a transition between stabilizable and non-stabilizable regimes.

Findings

High density measures are non-stabilizable.

The thermodynamic limit of recurrent configurations is a maximal stabilizable measure.

Self-organized criticality corresponds to a transition point between stabilizable and non-stabilizable states.

Abstract

We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable