In this paper we consider the Gibbs Sampler dynamics with simulated annealing and partially parallel updating scheme, as proposed by Trouv\'e in [13]. It is known that in some cases the support of the limiting measure does not coincide with the set of global maxima of the underlying energy function. We provide some new simple examples of this undesirable behavior. But then we prove that for one-dimensional models with nearest neighbor interaction the algorithm works ``generically''. We prove also that for 2 dimensional nearest neighbor ferromagnetic uniform interactions the algorithm works.
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