Potential Analysis

, 30:1 | Cite as

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

Open Access
Article

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 (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of \(n=\omega_d r^d\) sites, where ωd 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(logr), while the outradius is at most r + O(rα) for any α > 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 = πr2 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, improving on bounds of Fey and Redig.

Keywords

Abelian sandpile Asymptotic shape Discrete Laplacian Divisible sandpile Growth model Internal diffusion limited aggregation Rotor-router model 

Mathematics Subject Classifications (2000)

Primary 60G50 Secondary 35R35 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA
  2. 2.Microsoft ResearchRedmondUSA

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