Limiting directions for random walks in classical affine Weyl groups

Abstract

Let $W$ be a finite Weyl group and $\tilde{W}$ the corresponding affine Weyl group. A random element of $\tilde{W}$ can be obtained as a reduced random walk on the alcoves of $\tilde{W}$. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of $|W|$ many directions. We compute these directions when $W$ is $B_n$, $C_n$ and $D_n$ and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types $B$ and $C$ in the affirmative and for type $D$ in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the $D^{\ast }$-TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.

arXiv comment45 pages, 9 figures, minor improvements, final version

References in corpus (7)

• Stationary distributions of multi-type totally asymmetric exclusion processes
• Matrix representation of the stationary measure for the multispecies TASEP
• On the Two Species Asymmetric Exclusion Process with Semi-Permeable Boundaries
• Recursive structures in the multispecies TASEP
• The shape of a random affine Weyl group element and random core partitions
• Correlations in the Multispecies TASEP and a Conjecture by Lam
• The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps