About relaxation time of finite generalized Metropolis algorithms

In 1999 Catoni determined the critical rate H3 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 H3 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.