Limit shape of the leaky Abelian sandpile model with multiple layers

TL;DR

This paper extends the Leaky Abelian Sandpile Model to multiple layers and dimensions, analyzing the resulting limit shapes which range from polytopes to ellipsoids depending on parameters, revealing deep geometric and probabilistic properties.

Contribution

It introduces a multi-layer, multi-dimensional generalization of LASM and characterizes the asymptotic limit shapes under various leakiness conditions, a novel extension of prior models.

Findings

Limit shape converges to a scaled shape as grains increase.

Limit shape becomes a polytope when leakiness tends to infinity.

Limit shape approaches an ellipsoid as leakiness tends to one.

Abstract

In this paper we study a triple generalization of the Leaky Abelian Sandpile Model (LASM) of Alevy and Mkrtchyan, originally analyzed in the case of the square lattice in dimension two. First, we work in any dimension. Second, each site can hold several different stacks of sand, one for each of a certain given number of different layers or colors. Third, when a stack of one color at a site topples, it can send sand not only to its nearest neighbors in equal amounts, but to all possible locations and colors, according to a fixed but arbitrary mass distribution. Stacks of different colors can topple according to different distributions and different leakiness parameters, however the toppling rule should be site-independent. We obtain three main results. First, in this generality, when the LASM is started with $N$ grains of sand in one color at the origin, the final stable configuration, after scaling down by $\log N$, converges to a limit shape as $N$ goes to infinity. Second, when the leakiness parameter converges to infinity and the toppling distribution has finite range, the limit shape converges to a polytope. Third, when the leakiness parameters converge to one, which means the leakiness disappears, the limit shape of the sandpile converges to an ellipsoid. From a technical point of view, we rely on a strong relation between the Green function for random walk and the shape of the sandpile. Finally, the limit shape exhibits interesting duality properties, which we also investigate.