## Duke Mathematical Journal

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

### Convergence of the Abelian sandpile

Wesley Pegden and Charles K. Smart

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00127094-2079677

**Mathematical Reviews number (MathSciNet)**

MR3039676

**Zentralblatt MATH identifier**

1283.60124

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 35R35: Free boundary problems

#### 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