Sandpile group of infinite graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912418314059776 |
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| author | Kalinin, Nikita Khramov, Vladislav |
| author_facet | Kalinin, Nikita Khramov, Vladislav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_05346 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sandpile group of infinite graphs Kalinin, Nikita Khramov, Vladislav Group Theory Mathematical Physics Combinatorics Dynamical Systems 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. |
| title | Sandpile group of infinite graphs |
| topic | Group Theory Mathematical Physics Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2305.05346 |