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

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.

Published at http://dx.doi.org/10.1214/14-AOP915 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• An Inhomogeneous Multispecies TASEP on a Ring
• Hecke group algebras as quotients of affine Hecke algebras at level 0
• Total positivity in loop groups II: Chevalley generators
• Stationary probability of the identity for the TASEP on a ring

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• An Inhomogeneous Multispecies TASEP on a Ring
• Interacting particle systems and random walks on Hecke algebras
• The size of $t$-cores and hook lengths of random cells in random partitions
• Multiline queues with spectral parameters
• The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps
• A product formula for the TASEP on a ring
• Bender--Knuth Billiards in Coxeter Groups
• Limiting directions for random walks in classical affine Weyl groups
• TASEP in any Weyl Group