Eigenvectors for a random walk on a hyperplane arrangement

TL;DR

This paper derives explicit eigenvectors for the transition matrix of a specific random walk on hyperplane arrangements, combining combinatorics and analysis of stationary distributions.

Contribution

It provides a novel explicit construction of eigenvectors for the random walk's transition matrix using combinatorial and analytical methods.

Findings

Explicit eigenvectors for the transition matrix are obtained.

The approach combines combinatorics of face lattices with stationary distribution analysis.

Results facilitate understanding of the spectral properties of the random walk.

Abstract

We find explicit eigenvectors for the transition matrix of a random walk due to Bidegare, Hanlon and Rockmore. This is accomplished by using Brown and Diaconis' analysis of its stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelfand and Varchenko.