The critical density of the Stochastic Sandpile Model

TL;DR

This paper proves that the critical density of the stochastic sandpile model is less than one in all dimensions and positive on any vertex-transitive graph, extending previous results and simplifying proofs.

Contribution

It generalizes the critical density result from one dimension to all dimensions and extends positivity to all vertex-transitive graphs with a simpler proof.

Findings

Critical density is less than one in all dimensions.

Critical density is strictly positive on any vertex-transitive graph.

Provides a simplified proof for the positivity result.

Abstract

We study the stochastic sandpile model on $\mathbb{Z}^d$ and demonstrate that the critical density is strictly less than one in all dimensions. This generalizes a previous result by Hoffman, Hu, Richey, and Rizzolo (2022), which was limited to the one-dimensional case. In addition, we show that the critical density is strictly positive on any vertex-transitive graph, extending the earlier result of Sidoravicius and Teixeira (2018) and providing a simpler proof.