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July 2015 The shape of a random affine Weyl group element and random core partitions
Thomas Lam
Ann. Probab. 43(4): 1643-1662 (July 2015). DOI: 10.1214/14-AOP915

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.

Citation

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Thomas Lam. "The shape of a random affine Weyl group element and random core partitions." Ann. Probab. 43 (4) 1643 - 1662, July 2015. https://doi.org/10.1214/14-AOP915

Information

Received: 1 December 2012; Revised: 1 January 2014; Published: July 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1320.60028
MathSciNet: MR3353811
Digital Object Identifier: 10.1214/14-AOP915

Subjects:
Primary: 60C05, 60J10

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 4 • July 2015
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