Some universal estimates for reversible Markov chains

TL;DR

This paper establishes universal bounds on the convergence rates of reversible Markov chains, linking entropy increase to mixing times and introducing a novel ultrametric structure for analyzing their equilibrium behavior.

Contribution

It introduces a new identity connecting entropy increase to convergence, and proposes a universal ultrametric partition framework for reversible Markov chains.

Findings

Total variation convergence estimates via entropy increase

Universal ultrametric partition structure on state space

Control of global convergence using entropy for uniform reversible chains

Abstract

We obtain universal estimates on the convergence to equilibrium and the times of coupling for continuous time irreducible reversible finite-state Markov chains, both in the total variation and in the L^2 norms. The estimates in total variation norm are obtained using a novel identity relating the convergence to equilibrium of a reversible Markov chain to the increase in the entropy of its one-dimensional distributions. In addition, we propose a universal way of defining the ultrametric partition structure on the state space of such Markov chains. Finally, for chains reversible with respect to the uniform measure, we show how the global convergence to equilibrium can be controlled using the entropy accumulated by the chain.