# 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…

## 55 Citations

### Apollonian structure in the Abelian sandpile

- Geology
- 2012

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

- Mathematics
- 2015

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

- MathematicsCommunications in Mathematical Physics
- 2019

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

- MathematicsAnnales de l’Institut Henri Poincaré D
- 2020

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

- Mathematics
- 2019

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

- MathematicsProceedings of the National Academy of Sciences
- 2019

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

- MathematicsThe Annals of Probability
- 2019

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

- MathematicsJournal d'Analyse Mathématique
- 2019

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

- MathematicsJournal d'Analyse Mathématique
- 2019

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

- MathematicsThe Annals of Probability
- 2021

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…

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