  Let $W$ be a finite Weyl group and ${\hat{W}}$ be the corresponding affine
Weyl group. We show that a large element in ${\hat{W}}$, randomly generated by
(reduced) multiplication by simple generators, almost surely has one of
$|W|$-specific shapes. Equivalently, a reduced random walk in the regions of
the affine Coxeter arrangement asymptotically approaches one of $|W|$-many
directions. The coordinates of this direction, together with the probabilities
of each direction can be calculated via a Markov chain on $W$. Our results,
applied to type $\tilde{A}_{n-1}$, show that a large random $n$-core obtained
from the natural growth process has a limiting shape which is a
piecewise-linear graph. In this case, our random process is a periodic analogue
of TASEP, and our limiting shapes can be compared with Rost's theorem on the
limiting shape of TASEP.
