In 1999 Catoni determined the critical rate $H_3$ for the relaxation time of generalized Metropolis algorithms, models for which the speed of convergence to equilibrium can be strongly influenced by the effects of a possible almost periodicity. We recover this result with the help of Dobrushin's coefficient and give characterizations of $H_3$ in terms of other ergodic constants. In particular, we prove that it also governs the large deviation behavior of the singular gap for a sufficiently large but finite number of iterations of the underlying kernel at low temperature.
"About relaxation time of finite generalized Metropolis algorithms." Ann. Appl. Probab. 12 (4) 1492 - 1515, November 2002. https://doi.org/10.1214/aoap/1037125871