Sandpile group of infinite graphs

TL;DR

This paper extends the concept of sandpile groups to infinite graphs, specifically $ ext{Z}^2$, by defining a group of recurrent states related to harmonic functions, providing new algebraic insights and examples.

Contribution

It introduces a unified algebraic framework for sandpile groups on infinite graphs like $ ext{Z}^2$, connecting them to harmonic functions and exploring their torsion properties.

Findings

The group $C( ext{Z}^2, S)$ is isomorphic to a group of $S^1$-valued discrete harmonic functions.

Examples show $C( ext{Z}^2, S)$ can have no torsion or all torsions.

A projective limit perspective is discussed.

Abstract

For a finite connected graph $G$ and a non-empty subset $S$ of its vertices thought of sinks, the so-called critical group (or sandpile group) $C(G, S)$ has been studied for a long time. We present a class of graphs where such an extension can be made in a unified way. Similar extension was made by Maes, C. and Redig, F. and Saada, E., but we propose a more algebraic point of view. Namely, consider a $C$-net $S\subset \mathbb Z^2$. We define a sandpile dynamics on $\mathbb Z^2$ with the set $S$ of sinks. For such a choice of sinks, a relaxation of any bounded state is well defined. This allows us to define a group $C(\mathbb Z^2, S)$ of recurrent states of this model. We show that $C(\mathbb Z^2, S)$ is isomorphic to a group of $S^1$-valued discrete harmonic functions on $\mathbb Z^2\setminus S$. Examples of $S$, for which $C(\mathbb Z^2, S)$ has no torsion or has all torsions, are provided. Pontryagin dual point of view is investigated. A discussion about perspectives of a sandpile group for $\mathbb Z^2$ as a projective limit concludes this work.