Structure of some sand pile model

TL;DR

This paper investigates the structure of the Sand Pile Model's lattice, revealing its self-similarity, and introduces an extension to infinite grains with efficient construction algorithms and recursive formulas.

Contribution

It provides a detailed analysis of the SPM lattice structure, including self-similarity properties and algorithms for finite and infinite cases, with new recursive formulas for lattice cardinality.

Findings

The SPM lattice exhibits strong self-similarity.

Efficient algorithms are developed for constructing SPM(n) and SPM(infini).

Recursive formulas for the size of SPM(n) are proposed.

Abstract

SPM (Sand Pile Model) is a simple discrete dynamical system used in physics to represent granular objects. It is deeply related to integer partitions, and many other combinatorics problems, such as tilings or rewriting systems. The evolution of the system started with n stacked grains generates a lattice, denoted by SPM(n). We study here the structure of this lattice. We first explain how it can be constructed, by showing its strong self-similarity property. Then, we define SPM(infini), a natural extension of SPM when one starts with an infinite number of grains. Again, we give an efficient construction algorithm and a coding of this lattice using a self-similar tree. The two approaches give different recursive formulae for the cardinal of SPM(n), where no closed formula have ever been found.