Multiple and inverse topplings in the Abelian Sandpile Model

TL;DR

This paper explores the Abelian Sandpile Model's dynamics, extending it with antitoppling rules to form a non-abelian monoid, and investigates the algebraic properties and implications of this larger structure.

Contribution

It introduces and analyzes a non-abelian monoid by including antitoppling rules in the Abelian Sandpile Model, expanding understanding of its algebraic structure.

Findings

The monoid with antitoppling rules is non-abelian.

Algebraic properties of the larger monoid are characterized.

Implications for the structure of the model are described.

Abstract

The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand. The fundamental moves defining the dynamics are encoded by the toppling rules. The transition monoid corresponding to this dynamics in the set of stable configurations is abelian, a property which seems at the basis of our understanding of the model. By including also antitoppling rules, we introduce and investigate a larger monoid, which is not abelian anymore. We prove a number of algebraic properties of this monoid, and describe their practical implications on the emerging structures of the model.