Cluster Expansions and Pirogov - Sinai Theory for Long Range Spin Systems

We investigate the low temperature phases of lattice spin systems with interactions of Kac type, that is interactions that are weak but long range in such a way that the total interaction of one spin with all the others is of order unity. In particular we develop a systematic approach to convergent low temperature expansions in situations where interactions are weak but long range. This leads to a reformulation of the model in terms of a generalized abstract Pirogov - Sinai model, that is a representation in terms of contours interacting through cluster fields. The main point of our approach is that all quantities in the contour representation satisfy estimates that are uniform in the range of the interaction and depend only on the overall interaction strength. The extension of the Pirogov - Sinai theory to such models developed in [M. Zahradnik, Cluster expansions of small contours in abstract Pirogov - Sinai models, Markov Processes Relat. Fields, 2002, v.8, N3, 383-441] allows then the investigation of the low-temperature phase diagram of these models