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… 
View PDF on arXiv
43 Citations
Highly Influential Citations
2
Background Citations
16
Methods Citations
4

Figures from this paper

43 Citations

Citation Type

Citation Type

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…
Perturbed Divisible Sandpiles and Quadrature Surfaces
The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a…
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
TLDR
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…
Laplacian growth, sandpiles, and scaling limits
Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 21 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
TLDR
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.
Random Walk: A Modern Introduction
TLDR
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.
Self-organized criticality
We show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales. The temporal ``fingerprint'' of the self-organized…
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 Equa-tions of Second Order
Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation…
...
1
2
3
...