Critical densities in sandpile models with quenched or annealed disorder

TL;DR

This paper clarifies the relationships between different critical densities in sandpile models, showing that some previously assumed equalities do not hold, and extends definitions to better understand these densities in finite and infinite volumes.

Contribution

It demonstrates that the stationary and transition densities differ in quenched sandpile models and clarifies the conditions under which auxiliary densities are equal to the transition density.

Findings

Stationary and transition densities are not equal in quenched models.

Threshold density can be equal to transition density in certain cases.

Critical activity density in infinite volume equals 1.

Abstract

We discuss various critical densities in sandpile models. The stationary density is the average expected height in the stationary state of a finite-volume model; the transition density is the critical point in the infinite-volume counterpart. These two critical densities were generally assumed to be equal, but this has turned out to be wrong for deterministic sandpile models. We show they are not equal in a quenched version of the Manna sandpile model either. In the literature, when the transition density is simulated, it is implicitly or explicitly assumed to be equal to either the so-called threshold density or the so-called critical activity density. We properly define these auxiliary densities, and prove that in certain cases, the threshold density is equal to the transition density. We extend the definition of the critical activity density to infinite volume, and prove that in the standard infinite volume sandpile, it is equal to 1. Our results should bring some order in the precise relations between the various densities.