Phase transitions in the LMP model

Abstract

The Pirogov–Sinai strategy outlined in Chap. 10 is implemented in this chapter. It requires three main estimates. The first one is the existence of a chemical potential which equilibrates the pressures in the plus and minus restricted ensembles (which is the content of Sect. 11.1). In Sect. 11.2 it is then shown that it is possible to factorize the contribution in the contours weights of the interior part of the contours and the part at the boundaries. The contribution of the latter is estimated in Sect. 11.3 (this is the second main estimate, called “the energy estimate”), the contribution of the former (which is the third main estimate called “the surface correction to the pressure”) is studied in Sect. 11.4. This is the most delicate part of the whole proof. It involves a cluster expansion to estimate the contribution of the other contours in the interior of the given one, leading to a new effective hamiltonian. By means of an interpolation procedure the surface corrections to the pressure are reduced to an estimate on the decay of correlations for the new effective hamiltonian in the plus (or minus) restricted ensemble. Such properties are next proved in Sects. 11.5 and 11.6, by extending the Dobrushin uniqueness theory of Sect. 3.3. This is done in Sect. 11.5 in a general context, while in Sect. 11.6 it is proved that the LMP model indeed verifies the assumptions stated in Sect. 11.5. Some more technical estimates are given in Sects. 11.7 and 11.8, while theorems on cluster expansion are recalled in Sect. 11.9 without proof. References can be found in Sect. 12.6.