On the abelianity of the stochastic sandpile model

TL;DR

This paper investigates a stochastic variant of the Abelian Sandpile Model, demonstrating that certain algebraic properties like operator commutativity persist despite randomness, and confirming that the stationary distribution remains unaffected by addition methods.

Contribution

It proves that even in a stochastic ASM, key properties such as operator commutativity and distribution independence are preserved, extending the understanding of the model's algebraic structure.

Findings

Operators still commute despite stochasticity

Stationary distribution is independent of addition method

Some algebraic properties of ASM are retained

Abstract

We consider a stochastic variant of the Abelian Sandpile Model (ASM) on a finite graph, introduced by Chan, Marckert and Selig. Even though it is a more general model, some nice properties still hold. We show that on a certain probability space, even if we lose the group structure due to topplings not being deterministic, some operators still commute. As a corollary, we show that the stationary distribution still does not depend on how sand grains are added onto the graph in our model, answering a conjecture of Selig.