Functions of random walks on hyperplane arrangements

TL;DR

This paper explores random walks on hyperplane arrangements and their subarrangements, providing a unified framework with explicit spectral properties, including new examples like walks on acyclic orientations of graphs.

Contribution

It introduces a uniform approach to analyze hyperplane walks on subarrangements, deriving eigenvalues and stationary distributions, and includes novel examples such as acyclic orientations.

Findings

Diagonalizable matrices with known eigenvalues

Explicit stationary distributions for various walks

Good convergence rates to stationarity

Abstract

Many seemingly disparate Markov chains are unified when viewed as random walks on the set of chambers of a hyperplane arrangement. These include the Tsetlin library of theoretical computer science and various shuffling schemes. If only selected features of the chains are of interest, then the mixing times may change. We study the behavior of hyperplane walks, viewed on a subarrangement of a hyperplane arrangement. These include many new examples, for instance a random walk on the set of acyclic orientations of a graph. All such walks can be treated in a uniform fashion, yielding diagonalizable matrices with known eigenvalues, stationary distribution and good rates of convergence to stationarity.