• DOI:10.1007/s11118-008-9104-6
• Corpus ID: 2227479

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

@article{Levine2008StrongSA,
  title={Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile},
  author={Lionel Levine and Yuval Peres},
  journal={Potential Analysis},
  year={2008},
  volume={30},
  pages={1-27}
}

• Lionel Levine, Y. Peres
• Published 5 April 2007
• Mathematics
• Potential Analysis

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.