Reversible Markov chains: variational representations and ordering

Abstract

This pedagogical document explains three variational representations that are useful when comparing the efficiencies of reversible Markov chains: (i) the Dirichlet form and the associated variational representations of the spectral gaps; (ii) a variational representation of the asymptotic variance of an ergodic average; and (iii) the conductance, and the equivalence of a non-zero conductance to a non-zero right spectral gap.