The Limit Shape of the Leaky Abelian Sandpile Model

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.

30 pages, 10 figures. To be published in International Mathematics Research Notices. The proof of Lemma 3.3 has been simplified and we have corrected several typos