Polymer gas approach to N - body lattice

We give a simple proof, based only on combinatorial arguments, of the Koteck y-Preiss condition for the convergence of the cluster expansion. Then we consider spin systems with long range, N body interactions. We prove directly, using polymer gas representation, that the pressure may be written in terms of an absolutely convergent series uniformely in the volume, when the interaction is summable in a suitable sense. We give also an estimate of this radius of convergence. In order to get the proof we use a method introduced by Cassandro and Olivieri in early 80's. We apply this method to various concrete examples.