A comparison between different cycle decompositions for Metropolis dynamics

TL;DR

This paper compares two cycle decomposition methods for Metropolis dynamics Markov chains, establishing their equivalence to unify different approaches in analyzing metastability.

Contribution

It introduces a proof of equivalence between two cycle decompositions in Metropolis dynamics, linking different analytical frameworks for metastability.

Findings

Proves the equivalence of two cycle decompositions

Provides a unified understanding of metastability analysis

Bridges different methodological approaches in Markov chain analysis

Abstract

In the last decades the problem of metastability has been attacked on rigorous grounds via many different approaches and techniques which are briefly reviewed in this paper. It is then useful to understand connections between different point of views. In view of this we consider irreducible, aperiodic and reversible Markov chains with exponentially small transition probabilities in the framework of Metropolis dynamics. We compare two different cycle decompositions and prove their equivalence.