Comparison Theory for Markov Chains on Different State Spaces and Application to Random Walk on Derangements

TL;DR

This paper develops a comparison theory for Markov chains on different state spaces, enabling spectral bounds transfer and applying it to analyze a random walk on derangements.

Contribution

It extends spectral comparison methods to chains on different state spaces, broadening analysis tools for Markov chains without symmetry.

Findings

Spectral bounds can be transferred between chains on different spaces.

The method simplifies analysis of chains lacking symmetry.

Applied to a random walk on derangements, providing new insights.

Abstract

Let $X_{t}$ and $Y_{t}$ be two Markov chains, on state spaces $\Omega \subset \hat{\Omega}$. In this paper, we discuss how to prove bounds on the spectrum of $X_{t}$ based on bounds on the spectrum of $Y_{t}$. This generalizes work of Diaconis, Saloff-Coste, Yuen and others on comparison of chains in the case $\Omega = \hat{\Omega}$. The main tool is the extension of functions from the smaller space to the larger, which allows comparison of the entire spectrum of the two chains. The theory is used to give quick analyses of several chains without symmetry. The main application is to a `random transposition' walk on derangements.