Critical Densities in Sandpile Models with Quenched or Annealed Disorder} \runtit{Critical densities in sandpile models

A. Fey, R. Meester

2015, v.21, Issue 1, 57-84

ABSTRACT

We discuss various critical densities in sandpile models. The {\em stationary density} is the average expected height in the stationary state of a nonparametric finite-volume model; the {\em transition density} is the critical point in the parametric infinite-volume counterpart. These two critical densities were generally assumed to be equal, but this has turned out not to be the case for deterministic sandpile models. We show they are not equal in a quenched version of the classical Manna sandpile model either.

In the literature, when the transition density is simulated, it is often implicitly or explicitly assumed to be equal to either the so-called {\em threshold density} or the so-called {\em critical activity density}, which we define below. We prove that in certain cases, the threshold density is indeed equal to the transition density.

We extend the definition of the critical activity density to infinite volume models, 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 critical densities.

Keywords: sandpile model, phase transition, Manna model, critical density, transition density, activity density, quenched and annealed disorder, self-organized criticality