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Stochastic Sandpiles with Uniform Toppling Rule on the Line

David Beck-Tiefenbach, Robin Kaiser

Abstract

We consider the stochastic sandpile model with uniform toppling rule on the integer line. During a uniform toppling, with probability $1/3$ one particle is sent to the right of the toppled vertex, with probability $1/3$ one particle is sent to the left, and with probability $1/3$ two particles are sent out, one to the right and one to the left. We calculate exactly the stationary distribution of the stochastic sandpile Markov chain with this toppling rule on finite, connected subsets of the integers, and show that the infinite volume limit exists and is equal to the Dirac measure of the full configuration. For this end, we analyze where the excess mass leaves the system, when stabilizing the full configuration plus one additional particle on finite, connected subsets of the integers.

Stochastic Sandpiles with Uniform Toppling Rule on the Line

Abstract

We consider the stochastic sandpile model with uniform toppling rule on the integer line. During a uniform toppling, with probability one particle is sent to the right of the toppled vertex, with probability one particle is sent to the left, and with probability two particles are sent out, one to the right and one to the left. We calculate exactly the stationary distribution of the stochastic sandpile Markov chain with this toppling rule on finite, connected subsets of the integers, and show that the infinite volume limit exists and is equal to the Dirac measure of the full configuration. For this end, we analyze where the excess mass leaves the system, when stabilizing the full configuration plus one additional particle on finite, connected subsets of the integers.
Paper Structure (11 sections, 10 theorems, 71 equations, 3 figures)

This paper contains 11 sections, 10 theorems, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Concentration Phenomenon of the Stationary Distribution
  5. Sandpile Gambler's Ruin
  6. Concentration on the Full Configuration
  7. Infinite Volume Limit
  8. Hole Probabilities
  9. Stationary Distribution on Vacant Configurations
  10. Proof of the Infinite Volume Limit
  11. Conclusion

Key Result

Theorem 1.1

Let for $a,b\in\mathbb{Z}$ with $a<0<b$ denote by $\mu^{\text{stoch}}_{a,b}$ the stationary distribution of the stochastic sandpile Markov chain with uniform toppling rule on the set $[[a,b]]:=\{a,a+1,...,b-1,b\}$, where the outside is identified as the sink.

Figures (3)

  • Figure 1: The graph illustrates the average density $\rho_{G}(p)$ for $p$-topplings at certain values of $p$, where $G=[[1,100]]^2$ is a box with $10000$ vertices. We can see that the density seems to be linearly decaying as we increase the value of $p$.
  • Figure 2: The figure depicts the height $3$ cluster of typical samples of the stationary distribution of stochastic sandpiles with $p$-topplings on a box of side length $200$ for the subcritical and supercritical regime, as well as for a value of $p$ close to the critical regime. The five largest clusters are highlighted in color, the remaining clusters are depicted in gray.
  • Figure 3: The graph shows the probability of finding a cluster of height $3$ vertices spanning from the right side of a box of side length $L=120$ to the left side of the box. Our simulations suggest that the phase transition occurs when the average density of height $3$ vertices is equal to the critical Bernoulli site percolation threshhold on $\mathbbm{Z}^2$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1: Stabilization is Well-Defined
  • proof
  • Definition 3.1: Sandpile Gambler's Ruin Probabilities
  • Lemma 3.1: Recurrence Relation
  • proof
  • Proposition 3.1: Explicit Solution of the Sandpile Gambler's Ruin
  • proof
  • Remark 3.1: Namesake of the Sandpile Gambler's Ruin
  • Lemma 3.2: Transition Probabilities to Full Configuration
  • ...and 17 more