Limiting directions for random walks in classical affine Weyl groups

Abstract

Let W be a finite Weyl group and W̃ the corresponding affine Weyl group. A random element of W̃ can be obtained as a reduced random walk on the alcoves of W̃⁠. 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 and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process (TASEP) called the D*-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.