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