## The Annals of Probability

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

Thomas Lam

#### 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.

#### Article information

Source
Ann. Probab., Volume 43, Number 4 (2015), 1643-1662.

Dates
Revised: January 2014
First available in Project Euclid: 3 June 2015

https://projecteuclid.org/euclid.aop/1433341316

Digital Object Identifier
doi:10.1214/14-AOP915

Mathematical Reviews number (MathSciNet)
MR3353811

Zentralblatt MATH identifier
1320.60028

#### Citation

Lam, Thomas. The shape of a random affine Weyl group element and random core partitions. Ann. Probab. 43 (2015), no. 4, 1643--1662. doi:10.1214/14-AOP915. https://projecteuclid.org/euclid.aop/1433341316

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