Duke Mathematical Journal

Convergence of the Abelian sandpile

Wesley Pegden and Charles K. Smart

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Abstract

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice Zd, 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. 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. 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


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