I will be discussing random walks on the chambers of hyperplane arrangements in R^n. The general idea is that a collection of hyperplanes carves the underlying space into a bunch of disjoint pieces called faces. The n-dimensional faces are called chambers, and there is a natural semigroup product on the faces with respect to which the chambers form a two-sided ideal. One can construct a random walk on the chambers in terms of repeated left-multiplication by randomly chosen faces. It turns out that a wide variety of natural Markov chains can be interpreted within this framework, and a surprising amount is known about the general theory of these hyperplane walks.

In this talk, I will provide a brief overview of the subject, including several examples and a survey of established results. I will then discuss how to explicitly recover the top eigenfunctions in terms of projections onto subarrangements and demonstrate how they can be used to obtain bounds on the mixing times.