Let W be a finite Weyl group and W^ be the corresponding affine
Weyl group. We show that a large element in W^, randomly generated by
(reduced) multiplication by simple generators, almost surely has one of
β£Wβ£-specific shapes. Equivalently, a reduced random walk in the regions of
the affine Coxeter arrangement asymptotically approaches one of β£Wβ£-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 A~nβ1β, 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.Comment: Published at http://dx.doi.org/10.1214/14-AOP915 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org