Cut-off for a Class of Hyperplane Arrangement Walks

Consider a real hyperplane arrangement and let C denote the collection of the occuring chambers. Bidigare, Hanlon and Rockmore introduced a Markov chain on C which is a generalization of some card shuffling models used in computer science, biology and card games: the famous Tsetlin library used in dynamic file maintenance and cache maintenance and the riffle shuffles are two important examples of hyperplane walks. A strong stationary argument gives the upper bound for the separation distance
mixing time for this Markov chain. A coupon collector argument will lead to the lower bound, which is discussed for the first time. I will try to explain both the geometric and the probabilistic techniques used in the problem.