The Abelian sandpile; a mathematical introduction

TL;DR

This paper provides a rigorous mathematical introduction to the abelian sandpile model, clarifying classical results, filling gaps in existing proofs, and bridging the gap between physics and mathematics perspectives.

Contribution

It offers a simplified, rigorous treatment of well-known results, includes new proofs, and serves as an accessible introduction to the abelian sandpile model.

Findings

Provides a rigorous mathematical framework for the abelian sandpile

Completes missing proofs from the physics literature

Introduces new proofs and clarifies classical results

Abstract

We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find complete proofs in the literature. The paper tries to fill the gap between the mathematics and the physics literature on this subject, and also presents some new proofs. It can also serve as an introduction to the model.