Elementary Bounds On Mixing Times for Decomposable Markov Chains

TL;DR

This paper establishes new bounds on the mixing times of decomposable finite-state reversible Markov chains by relating them to the mixing properties of their projection and restriction chains, improving existing bounds in certain cases.

Contribution

It provides simple, improved bounds on mixing times for decomposable Markov chains using recent hitting and mixing time results, with illustrative examples.

Findings

Bounds relate original chain's mixing time to projection and restriction chains.

Results improve upon existing bounds in specific scenarios.

Examples demonstrate the effectiveness of the new bounds.

Abstract

Many finite-state reversible Markov chains can be naturally decomposed into "projection" and "restriction" chains. In this paper we provide bounds on the total variation mixing times of the original chain in terms of the mixing properties of these related chains. This paper is in the tradition of existing bounds on Poincare and log-Sobolev constants of Markov chains in terms of similar decompositions [JSTV04, MR02, MR06, MY09]. Our proofs are simple, relying largely on recent results relating hitting and mixing times of reversible Markov chains [PS13, Oli12]. We describe situations in which our results give substantially better bounds than those obtained by applying existing decomposition results and provide examples for illustration.