## Duke Mathematical Journal

### Convergence of the Abelian sandpile

#### Abstract

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 topple, distributing one 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 been proved about the appearance of the stable configurations. We use partial differential equation techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n\to\infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

#### Article information

Source
Duke Math. J., Volume 162, Number 4 (2013), 627-642.

Dates
First available in Project Euclid: 15 March 2013

https://projecteuclid.org/euclid.dmj/1363355689

Digital Object Identifier
doi:10.1215/00127094-2079677

Mathematical Reviews number (MathSciNet)
MR3039676

Zentralblatt MATH identifier
1283.60124

#### Citation

Pegden, Wesley; Smart, Charles K. Convergence of the Abelian sandpile. Duke Math. J. 162 (2013), no. 4, 627--642. doi:10.1215/00127094-2079677. https://projecteuclid.org/euclid.dmj/1363355689

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