Decay of correlation in simultaniously existing phases of Pirogov-Sinai model via a modification of Dobrushin's uniqueness method

The decay of correlation mentioned in the title is an essential information needed in order to control the phases of some Kac-systems. In the work of A. Mazel, J.Lebowitz and E. Presutti it was shown rigorously the existence of a phase transition with the density as order parameter for these systems. Although, one has not a unique state we were nevertheless able to modify Drobushin's uniqueness method in such a way, that it becomes applicable. The Dobrushin condition cannot hold for all configurations, because otherwise it would imply global uniqueness. However, the configurations which violate the condition have small probability in the phase under consideration. This can be formalized using the notion of restricted ensemble. The decay of correlation for the restricted ensemble can be used to control the finite volume corrections of the pressure. The latter one are essential to obtain information about the initially considered systems from the restricted ensemble.