The Limit Shape of the Leaky Abelian Sandpile Model

TL;DR

This paper analyzes the limit shape of the leaky abelian sandpile model, showing it transitions from a circle to a diamond as the leak parameter varies, and establishes convergence properties related to the model's parameters.

Contribution

The paper computes the limit shape of the Leaky-ASM as a function of the leak parameter and connects it to random walk probabilities, extending understanding of growth models.

Findings

Limit shape converges to a circle as leak parameter approaches 1.

Limit shape becomes a diamond as leak parameter approaches infinity.

Convergence to the circle shape occurs when the leak parameter approaches 1 slower than any power of n.

Abstract

The leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in $\mathbb{Z}^2$ and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. We compute the limit shape as a function of $d$ in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as $d\to 1$ and a diamond as $d\to\infty$. We compute the limit shape by comparing the odometer function at a site to the probability that a killed random walk dies at that site. When $d\to 1$ the Leaky-ASM converges to the abelian sandpile model (ASM) with a modified initial configuration. We also prove the limit shape is a circle when simultaneously with $n\to\infty$ we have that $d=d_n$ converges to $1$ slower than any power of $n$. To gain information about the ASM faster convergence is necessary.