We prove that the Markov operator corresponding to the two-variable, non-reversible Gibbs sampler has spectrum which is entirely real and non-negative, thus providing a first step towards the spectral analysis of MCMC algorithms in the non-reversible case. We also provide an extension to Metropolis-Hastings components, and connect the spectrum of an algorithm to the spectrum of its marginal chain.
"Spectral bounds for certain two-factor non-reversible MCMC algorithms." Electron. Commun. Probab. 20 1 - 10, 2015. https://doi.org/10.1214/ECP.v20-4528