Anton Bovier, Milos Zahradnik
Cluster Expansions and Pirogov Sinai Theory 
for Long Range Spin Systems 
(420K, postscript file)

ABSTRACT.  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 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 [Z3, see the next 
contribution to the archive] allows then the investigation of the low-temperature phase diagram of these models.