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=ωdrd
sites, where ωd is the volume of the unit ball in Rd, 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/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.Comment: [v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian
sandpile. [v4] Added references and improved exposition in sections 2 and 4.
[v5] Final version, to appear in Potential Analysi