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 = ωdr d sites, where ωd is the volume of the unit ball in 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 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 / √ 3, and the outradius is at most (r + o(r)) / √ 2. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.
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