THE SHAPE OF A RANDOM AFFINE WEYL GROUP ELEMENT AND RANDOM CORE PARTITIONS

Let W be a finite Weyl group and (W) over cap be the corresponding affine Weyl group. We show that a large element in (W) over cap, randomly generated by (reduced) multiplication by simple generators, almost surely has one of vertical bar W vertical bar-specific shapes. Equivalently, a reduced random walk in the regions of the affine Coxeter arrangement asymptotically approaches one of vertical bar W vertical bar-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) over tilde (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.