Stabilizability and percolation in the infinite volume sandpile model

TL;DR

This paper investigates the conditions under which initial configurations in the infinite volume sandpile model are stabilizable, analyzing the influence of density and percolation properties of toppled sites across different dimensions.

Contribution

It establishes that stabilizability is procedure-independent and characterizes the percolation behavior of toppled sites for various densities in the model.

Findings

Stabilizability does not depend on the stabilization procedure.

Configurations with density 1 in 1D are not stabilizable.

For small densities, toppled sites form subcritical clusters with exponential decay.

Abstract

We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $\mu$, are $\mu$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $\mu$ a product measure with density $\rho=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $\mu$ is not stabilizable. Furthermore, we study, for values of $\rho$ such that $\mu$ is stabilizable, percolation of toppled sites. We find that for $\rho>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.