Abelian Sandpiles on Cylinders

TL;DR

This paper investigates a variant of the Abelian Sandpile Model on cylindrical geometries, revealing a unique ladder structure in avalanche size distribution when the circumference is much smaller than the width.

Contribution

It introduces a novel geometric setting for the Abelian Sandpile Model and uncovers a previously unobserved ladder structure in avalanche size probabilities.

Findings

Avalanche size distribution exhibits a ladder structure on cylinders with small circumference.

The first ladder step includes avalanches up to size w c/2, nearly equally probable.

An exponential tail of order about 10c characterizes larger avalanches.

Abstract

We study here a variant of the Abelian Sandpile Model, where the playground is a cylinder of width $w$ and of circumference c. When c << w, we describe a phenomenon which has not been observed in other geometries: the probability distribution of avalanche sizes has a ladder structure, with the first step consisting of avalanches of size up to w c/2 that are essentially equiprobable, except for a small exponential tail of order about 10c. We explain this phenomenon and describe subsequent steps.