Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees
The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behav...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2008-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3618/pdf |
| _version_ | 1797270416911237120 |
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| author | Itamar Landau Lionel Levine Yuval Peres |
| author_facet | Itamar Landau Lionel Levine Yuval Peres |
| author_sort | Itamar Landau |
| collection | DOAJ |
| description | The sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball. |
| first_indexed | 2024-04-25T02:03:56Z |
| format | Article |
| id | doaj.art-5b2d97284334472891b1f0c41dec3e56 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:56Z |
| publishDate | 2008-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-5b2d97284334472891b1f0c41dec3e562024-03-07T14:38:06ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502008-01-01DMTCS Proceedings vol. AJ,...Proceedings10.46298/dmtcs.36183618Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on TreesItamar Landau0Lionel Levine1Yuval Peres2Department of Mathematics [Berkeley]Department of Mathematics [Berkeley]Theory Group - Microsoft ResearchThe sandpile group of a graph $G$ is an abelian group whose order is the number of spanning trees of $G$. We find the decomposition of the sandpile group into cyclic subgroups when $G$ is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice $\mathbb{Z}^2$. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.https://dmtcs.episciences.org/3618/pdfabelian sandpilechip-firing gameregular treerotor-router modelsandpile group[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][info.info-hc] computer science [cs]/human-computer interaction [cs.hc] |
| spellingShingle | Itamar Landau Lionel Levine Yuval Peres Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees Discrete Mathematics & Theoretical Computer Science abelian sandpile chip-firing game regular tree rotor-router model sandpile group [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [info.info-hc] computer science [cs]/human-computer interaction [cs.hc] |
| title | Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees |
| title_full | Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees |
| title_fullStr | Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees |
| title_full_unstemmed | Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees |
| title_short | Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees |
| title_sort | chip firing and rotor routing on mathbb z d and on trees |
| topic | abelian sandpile chip-firing game regular tree rotor-router model sandpile group [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [info.info-hc] computer science [cs]/human-computer interaction [cs.hc] |
| url | https://dmtcs.episciences.org/3618/pdf |
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