Convergence of the Abelian sandpile

@article{Pegden2011ConvergenceOT,
  title={Convergence of the Abelian sandpile},
  author={Wesley Pegden and Charles K. Smart},
  journal={Duke Mathematical Journal},
  year={2011},
  volume={162},
  pages={627-642}
}
The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$. However, little has… 

Figures from this paper

Apollonian structure in the Abelian sandpile

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun

Tropical curves in sandpile models

A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to

Sandpiles on the Square Lattice

AbstractWe give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $${\mathbb{Z}^2}$$Z2 . We also determine the asymptotic spectral gap,

Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner

Sandpile dynamics on periodic tiling graphs

Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A method of computing

Harmonic dynamics of the abelian sandpile

It is demonstrated that the self-similar fractal structures arising in the abelian sandpile show smooth dynamics under harmonic fields, similar to sand dunes which travel, transform, and merge, depending on the wind, and that the existence of several scaling limits for infinite domains is conjecture.

Cut-off for sandpiles on tiling graphs

Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A general method of

Discrete Balayage and Boundary Sandpile

We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on ℤd (d ≥ 2) onto the boundary of an (a

Discrete Balayage and Boundary Sandpile

We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on ℤd (d ≥ 2) onto the boundary of an (a

Convergence of the random Abelian sandpile

We prove that Abelian sandpiles with random initial states converge almost surely to unique scaling limits. The proof follows the Armstrong-Smart program for stochastic homogenization of uniformly
...

References

SHOWING 1-10 OF 23 REFERENCES

Patterns formed by addition of grains to only one site of an abelian sandpile

Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ℤd. Any site with at least 2d particles then topples by sending one

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is

WHAT IS a sandpile ?

An abelian sandpile is a collection of indistinguishable chips distributed among the vertices of a graph. More precisely, it is a function from the vertices to the nonnegative integers, indicating

Self-organized critical state of sandpile automaton models.

  • Dhar
  • Computer Science, Physics
    Physical review letters
  • 1990
The critical state is characterized, and its entropy for an arbitrary finite lattice in any dimension is determined, and the two-point correlation function is shown to satisfy a linear equation.

Self-organized criticality

The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box

Random Walk: A Modern Introduction

This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice and is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.

Weak convergence methods for nonlinear partial differential equations

As working analysts we do not have to be reminded about the significance of understanding weak convergence. The routine of our work, the pattern of our daily lives, consists in searching for

Weak convergence methods for nonlinear partial differential equations

In this report, I have collected the proofs that Professor Gantumur Tsogtgerel, Dr. Brian Seguin, Benjamin Landon and I have developed in the summer of 2012 while studying various weak convergence

Elliptic Partial Differential Equations of Second Order

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations