The shape of a random affine Weyl group element and random core partitions

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 |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 ~An.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.