Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

Abstract

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work. For the shape consisting of $n={\omega }_d{r}^d$ sites, where ${\omega }_d$ is the volume of the unit ball in ${\mathbb{R}}^d$, we show that the inradius of the set of occupied sites is at least $r-O(\log r)$, while the outradius is at most $r+O(r^{\alpha })$ for any $\alpha >1-1/d$. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with $n=\pi r^2$ particles, we show that the inradius is at least $r/\sqrt{3}$, and the outradius is at most $(r+o(r))/\sqrt{2}$. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions.

[v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian sandpile. [v4] Added references and improved exposition in sections 2 and 4. [v5] Final version, to appear in Potential Analysis

References in corpus (3)

• Scaling Limits for Internal Aggregation Models with Multiple Sources
• Limiting shapes for deterministic centrally seeded growth models
• The Rotor-Router Model

Cited by in corpus (43)

• Chip-Firing and Rotor-Routing on Directed Graphs
• Scaling Limits for Internal Aggregation Models with Multiple Sources
• Growth Rates and Explosions in Sandpiles
• Convergence of the Abelian sandpile
• The Rotor-Router Model on Regular Trees
• Internal DLA and the Gaussian free field
• From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models
• Limiting shapes for deterministic centrally seeded growth models
• The divisible sandpile at critical density
• Fast simulation of large-scale growth models
• Pattern formation in fast-growing sandpiles
• A sandpile model for proportionate growth
• Discrete low-discrepancy sequences
• Sandpile solitons via smoothing of superharmonic functions
• Exact sampling and fast mixing of Activated Random Walk
• Limiting shapes for a non-abelian sandpile growth model and related cellular automata
• Convergence of the random Abelian sandpile
• Internal DLA on Sierpinski gasket graphs
• Modelling proportionate growth
• Proportionate growth in patterns formed in the rotor-router model
• Internal Diffusion-Limited aggregation with uniform starting points
• A note on fluctuations for internal diffusion limited aggregation
• The Cover Time of Deterministic Random Walks
• Spiral Structures in the Rotor-Router Walk
• Rotor walks on transient graphs and the wired spanning forest
• Rotor-Router Walk on a Semi-infinite Cylinder
• Discrete analogue computing with rotor-routers
• Escape rates for rotor walk in Z^d
• Diamond Aggregation
• Scaling limit for a long-range divisible sandpile
• Euler tours and unicycles in the rotor-router model
• Abelian logic gates
• Rotor walks on general trees
• A rotor configuration in Z^d where Schramm's bound of escape rates attains
• Internal Aggregation Models on Comb Lattices
• Rotor-router aggregation on the layered square lattice
• Odometers of Divisible Sandpile Models: Scaling Limits, iDLA and Obstacle Problems. A Survey
• Partial Balayage and a Generalization of the Divisible Sandpile Model
• Perfect boundaries in rotor-router aggregation on cylinders
• The concentration inequality for a discrete height function model perturbed by random potential
• Asymptotics of the odometer function for the internal Diffusion Limited Aggregation model
• Formation of an interface by competitive erosion
• Notes on identical configurations in Abelian Sandpile Model with initial height