About relaxation time of finite generalized metropolis algorithms

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.