The Limit Shape of the Leaky Abelian Sandpile Model

Abstract

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 ASM with a modified initial configuration. We also prove that 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.