Stochastic Sandpile model: exact sampling and complete graph

2025/07/02 by Campailla, Concetta, Forien, Nicolas
#60J27 #60K35 #82C22 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.01572

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.

Citations

• The critical density of the Stochastic Sandpile Model
• The density conjecture for activated random walk
• Universality Conjectures for Activated Random Walk
• The number of particles in activated random walk on the complete graph
• Active Phase for the Stochastic Sandpile on Z
• Exact sampling and fast mixing of Activated Random Walk
• Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks
• Critical densities in sandpile models with quenched or annealed disorder

Related

• Sandpile models
• A natural stochastic extension of the sandpile model on a graph
• On the abelianity of the stochastic sandpile model
• Abelian and stochastic sandpile models on complete bipartite graphs
• The critical density of the Stochastic Sandpile Model