The critical density of the Stochastic Sandpile Model

TL;DR

The paper resolves a central question about the stochastic sandpile model by proving that the critical density on $oldsymbol{\mathbb{Z}^d}$ is strictly less than one in every dimension and that the critical density is strictly positive on any vertex-transitive graph. The authors develop a Diaconis-Fulton graphical framework, introduce weak stabilisation, and construct a stochastic comparison with ARWD to transfer activation results from ARW-type processes to SSM. A key component is the ARWD-SSM coupling, augmented by a multiscale dormitory hierarchy and a sleep-mask strategy that yields exponential stabilisation times on the torus, which in turn imply $ u_c<1$. The positive lower bound on $ u_c$ on vertex-transitive graphs is established via ghost particles, extending prior one-dimensional results and offering a simpler proof. Together, these results advance the rigorous understanding of the SSM and its relation to ARW universality, with the stochastic comparison framework potentially applicable to broader non-abelian particle systems.

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.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1
• Lemma 2.1: Lemma 4 in RS
• Definition 2.1
• Definition 2.2
• Lemma 2.2
• proof