I will be discussing random walks on the chambers of (finite, central) arrangements of hyperplanes in $\mathbf{R}^n$. The general idea is that a collection of hyperplanes carves the underlying space into a bunch of 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. By endowing the set of faces with a probability measure, one can construct a random walk on the chambers in terms of repeated left-multiplication by randomly chosen faces. One of the reasons that these hyperplane chamber walks are so interesting is that a wide variety of Markov chains can be interpreted within this framework. Moreover, the general theory is largely understood.

In this talk, I will provide a brief overview of the subject, including several examples and a survey of the known results. I will then discuss how to explicitly recover some of the associated eigenfunctions in terms of projections onto subarrangements and demonstrate various uses for these objects.

The talk will be aimed at a general audience.