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Stochastic Sandpile Model: exact sampling and complete graph

Concetta Campailla, Nicolas Forien

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.

Stochastic Sandpile Model: exact sampling and complete graph

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 . 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.

Paper Structure

This paper contains 21 sections, 12 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and main results
  3. Definition of the driven-dissipative Markov chain
  4. Stationary density on the complete graph
  5. Half-toppling dynamics
  6. Exact sampling result
  7. A new point of view on half-topplings
  8. Outline of the paper
  9. Comments and perspectives
  10. The model with generic instability threshold
  11. An alternative strategy based on exact sampling
  12. Critical window
  13. Proof of Theorem \ref{['thmstationarydist']}: exact sampling
  14. The microscopic chain of sleeping/active particles
  15. Proof of the upper bound in Theorem \ref{['mainthm2']}
  16. ...and 6 more sections

Key Result

Theorem 2.1

For any $\varepsilon>0$, there exists $c>0$ such that, for $N$ large enough, where In particular, $\pi_N$ tends to the Dirac measure at $\rho_s$ when $N\to\infty$.

Figures (1)

  • Figure 1: The two different stabilisation procedures of the configuration $(\eta_0+\mathds{1}_{x^*},h_0)$ used in the proof of Theorem \ref{['thmstationarydist']}.

Theorems & Definitions (21)

  • Theorem 2.1
  • Lemma 2.2: Abelian property
  • Theorem 2.3
  • Lemma 2.4
  • Definition 5.1
  • Lemma 5.2
  • proof
  • Lemma 6.1
  • Lemma 6.2
  • Lemma 6.3
  • ...and 11 more