Structure and eigenvalues of heat-bath Markov chains

TL;DR

This paper proves that heat-bath Markov chains have no negative eigenvalues, impacting their analysis and applications in spin systems and contingency table sampling, with implications for mixing time analysis.

Contribution

It establishes a general proof that heat-bath chains lack negative eigenvalues and offers new characterizations and potential generalizations.

Findings

Heat-bath chains have no negative eigenvalues.

Applications to spin systems and contingency tables.

Implications for mixing time analysis.

Abstract

We prove that heat-bath chains (which we define in a general setting) have no negative eigenvalues. Two applications of this result are presented: one to single-site heat-bath chains for spin systems and one to a heat-bath Markov chain for sampling contingency tables. Some implications of our main result for the analysis of the mixing time of heat-bath Markov chains are discussed. We also prove an alternative characterisation of heat-bath chains, and consider possible generalisations.