Abstract
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.
Highlights
Chip-Firing and Rotor-Routing on ZdRotor-router walk is a deterministic analogue of random walk, first studied by Priezzhev et al [22] under the name “Eulerian walkers.” At each site in the integer lattice Z2 is a rotor pointing north, south, east or west
The sandpile group of a graph G is an abelian group whose order is the number of spanning trees of G
We summarize recent progress in understanding the shapes of two combinatorial growth models, the rotor-router model and the chip-firing or abelian sandpile model
Summary
Rotor-router walk is a deterministic analogue of random walk, first studied by Priezzhev et al [22] under the name “Eulerian walkers.” At each site in the integer lattice Z2 is a rotor pointing north, south, east or west. Let An denote the resulting region of n occupied sites. Fey and Redig [10] show that An contains a diamond These results do not rule out the possibility of “holes” in An far from the boundary or of long tendrils extending far beyond the boundary of the ball, provided the volume of these features is negligible compared to n. Theorem 1.1 [18] Let An be the region formed by rotor-router aggregation in Zd starting from n particles at the origin and any initial rotor state. Theorem 1.2 [18] Let Sn be the set of sites that are visited by the classical abelian sandpile model in Zd, starting from n particles at the origin. To our knowledge even the existence of an asymptotic shape is not known
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
