Limiting shapes for deterministic centrally seeded growth models

Abstract

We study the rotor router model and two deterministic sandpile models. For the rotor router model in ${\mathbb{Z}}^d$, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter $h$ (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension $d\ge 1$. For the rotor router model, the limiting shape is a sphere for all values of $h$. For one of the sandpile models, and $h=2d-2$ (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit $h\to -\infty$. Finally, we prove that the rotor router shape contains a diamond.

18 pages, 3 figures, some errors corrected and more explanation added, to appear in Journal of Statistical Physics

References in corpus (7)

• Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
• Introduction to the Sandpile Model
• Spherical Asymptotics for the Rotor-Router Model in Z^d
• The Abelian sandpile; a mathematical introduction
• Dynamics of Eulerian walkers
• The Rotor-Router Model
• A deterministic sandpile automaton revisited

Cited by in corpus (14)

• Chip-Firing and Rotor-Routing on Directed Graphs
• Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
• Growth Rates and Explosions in Sandpiles
• Pattern formation in growing sandpiles
• Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks
• A sandpile model for proportionate growth
• Pattern formation in fast-growing sandpiles
• Effect of Noise on Patterns Formed by Growing Sandpiles
• Asymptotic shape of the region visited by an Eulerian Walker
• A growth model based on the arithmetic $Z$-game
• Limit Theorems for Internal Aggregation Models
• Modelling proportionate growth
• Limiting shapes for a non-abelian sandpile growth model and related cellular automata
• A note on the abelian sandpile in Z^d