Sandpile group of infinite graphs
Nikita Kalinin, Vladislav Khramov

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 and a non-empty subset of its vertices thought of sinks, the so-called critical group (or sandpile group) 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 -net . We define a sandpile dynamics on with the set of sinks. For such a choice of sinks, a relaxation of any bounded state is well defined. This allows us to define a group of recurrent states of this model. We show that is isomorphic to a group of -valued discrete harmonic functions on . Examples of , for which has no torsion or has all torsions, are…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Theoretical and Computational Physics · Mathematical Dynamics and Fractals
