This paper offers a personal review of some things we’ve learned about rates of convergence of Markov chains to their stationary distributions. The main topic is ways of speeding up diffusive behavior. It also points to open problems and how much more there is to do.
"Some things we’ve learned (about Markov chain Monte Carlo)." Bernoulli 19 (4) 1294 - 1305, September 2013. https://doi.org/10.3150/12-BEJSP09