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 natural generalization of famous card shuffling models used in computer science, biology and card games. In this talk, we will discuss the mixing time of these walks with respect to the separation distance, with a special focus on lower bounds. I will try to explain both the geometric and the probabilistic techniques used in the problem.