Chip-Firing and Rotor-Routing on $\mathbb{Z}^d$ and on Trees
Abstrak
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.
Topik & Kata Kunci
Penulis (3)
Itamar Landau
Lionel Levine
Yuval Peres
Akses Cepat
- Tahun Terbit
- 2008
- Sumber Database
- DOAJ
- DOI
- 10.46298/dmtcs.3618
- Akses
- Open Access ✓