Random walks on the chambers of a hyperplane arrangement

Abstract

The Bidigare-Hanlon-Rockmore random walk on the chambers of a real hyperplane arrangement is a Markov chain that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We will introduce lower bounds for the separation distance and a strong stationary time, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. We will also discuss how the method for the lower bound can be used to prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system, such as the Ising model, reproving a result of Ding and Peres.