Convergence of the Abelian sandpile

TL;DR

This paper proves that the rescaled stable configurations of the Abelian sandpile model converge to a unique fractal pattern, using PDE techniques to characterize the limit as a solution to an elliptic obstacle problem.

Contribution

It establishes the convergence of rescaled stable configurations to a unique limit and characterizes this limit via PDE methods, advancing understanding of the model's asymptotic behavior.

Findings

Rescaled stable configurations converge to a unique fractal pattern.

The limit is characterized as the Laplacian of an elliptic obstacle problem.

The approach uses PDE techniques to analyze a discrete growth model.

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 {\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 been proved about the appearance of the stable configurations. We use PDE 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.