  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 $n=\omega_d r^d$
sites, where $\omega_d$ 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 $r+O(r^\alpha)$ for any $\alpha > 1-1/d$. 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 $n=\pi
r^2$ 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.
