Stochastic Sandpile Model: exact sampling and complete graph

TL;DR

This paper presents an exact sampling method for the stationary distribution of the Stochastic Sandpile Model on finite graphs and analyzes its behavior on complete graphs as the number of vertices grows large.

Contribution

It introduces a procedure for exact sampling of the model's stationary distribution on all connected finite graphs and studies the model's asymptotic density on complete graphs.

Findings

Exact sampling procedure for finite graphs

Stationary density approaches 1/2 on large complete graphs

New perspective on active and sleeping particles in the model

Abstract

We study the dynamics of the Stochastic Sandpile Model on finite graphs, with two main results. First, we describe a procedure to exactly sample from the stationary distribution of the model in all connected finite graphs, extending a result obtained by Levine and Liang for Activated Random Walks. Then, we study the model on the complete graph with a number of vertices tending to infinity and show that the stationary density tends to $1/2$. Along the way, we introduce a new point of view on the dynamics of the model, with active and sleeping particles, which may be of independent interest.