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

TL;DR

This paper studies the asymptotic shape of large elements in affine Weyl groups and random core partitions, revealing that they tend to specific directions and shapes, with connections to Markov chains and TASEP.

Contribution

It introduces a probabilistic framework for understanding the limiting shapes of affine Weyl group elements and random core partitions, linking combinatorics, algebra, and stochastic processes.

Findings

Large affine Weyl group elements have specific asymptotic shapes.

Random walks in affine Coxeter arrangements approach particular directions.

Limiting shapes of random n-cores resemble piecewise-linear graphs, related to TASEP.

Abstract

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.