I will talk about random infinite reduced words in Coxeter groups, or equivalently walks in the chambers of the Coxeter hyperplane arrangement subject to the condition that each hyperplane is crossed at most once. In the case of affine Coxeter groups (those acting on Euclidean geometries) this random process has a nice limiting behavior.
In the case of the affine Coxeter group for SL_n, this random process is related to: random growth models on "core" partitions, multi-type TASEP on a circle, and a periodic version of TASEP on a line where some particles are constrained to jump simultaneously.