# Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement

## TL;DR

This paper investigates Markov chains on hyperplane arrangement chambers, providing exact formulas and bounds for mixing times, cutoff phenomena, and extending results to Glauber dynamics, thus generalizing classical shuffling models.

## Contribution

It introduces new bounds and formulas for mixing times and cutoff phenomena in hyperplane arrangement walks, extending classical models like the Tsetlin library.

## Key findings

- Exact formula for separation distance in hyperplane arrangement walk
- Lower bounds for mixing times under certain conditions
- Proved a uniform lower bound for Glauber dynamics mixing time

## Abstract

This paper studies Markov chains on the chambers of real hyperplane arrangements, a model that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We discuss cutoff for the Tsetlin library for general weights, and we give an exact formula for the separation distance for the hyperplane arrangement walk. We introduce lower bounds, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. Using similar techniques, we also prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system.