Polymer gas approach to N-body lattice systems

We give a simple proof, based only on combinatorial arguments, of the Kotec ky-Preiss condition for the convergence of the cluster expansion. Then we c onsider spin systems with long-range N-body interactions. We prove directly , using the polymer gas representation, that the pressure may be written in terms of an absolutely convergent series uniformly in the volume when the interaction is summable in a suitable sense. We also give an estimate of th is radius of convergence. In order to get the proof we use a method introdu ced by Cassandro and Olivieri in the early 1980s. We apply this method to v arious concrete examples.