Approaching criticality via the zero dissipation limit in the abelian avalanche model

TL;DR

This paper studies a continuous height abelian sandpile model with small dissipation, proving the existence of a stationary measure, its non-critical behavior for non-zero dissipation, and the recovery of criticality in the zero dissipation limit.

Contribution

It provides a rigorous analysis of the zero dissipation limit in the abelian avalanche model, connecting non-critical and critical regimes with precise bounds and exponents.

Findings

Stationary measure exists for non-zero dissipation.

Model exhibits exponential decay of correlations away from criticality.

Critical behavior is recovered in the zero dissipation limit, with mean-field exponents for high dimensions.

Abstract

The discrete height abelian sandpile model was introduced by Bak, Tang & Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for $d > 4$.