Random walks on hyperplane arrangements and stopping times

TL;DR

This paper studies a Markov chain on chambers of hyperplane arrangements, extending card shuffling models, and provides explicit bounds for its convergence to stationarity using strong stationary arguments.

Contribution

It introduces strong stationary arguments for the hyperplane arrangement Markov chain, offering explicit bounds for the separation distance and generalizing existing shuffling models.

Findings

Established explicit bounds for the separation distance.

Extended card shuffling models to hyperplane arrangements.

Provided strong stationary arguments for the Markov chain.

Abstract

Consider a real hyperplane arrangement and let $\mathcal{C}$ denote the occurring chambers. Bidigare, Hanlon and Rockmore introduced a Markov chain on $\mathcal{C}$ which is a generalization of some card shuffling models used in computer science, biology and card games. This paper introduces strong stationary arguments for this Markov chain, which provide explicit bounds for the separation distance.