On the abelianity of the stochastic sandpile model

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.