Abstract
We give a simple proof, based only on combinatorial arguments, of the Kotecký–Preiss condition for the convergence of the cluster expansion. Then we consider 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 this radius of convergence. In order to get the proof we use a method introduced by Cassandro and Olivieri in the early 1980s. We apply this method to various concrete examples.
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Procacci, A., Scoppola, B. Polymer Gas Approach to N-Body Lattice Systems. Journal of Statistical Physics 96, 49–68 (1999). https://doi.org/10.1023/A:1004564214528
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DOI: https://doi.org/10.1023/A:1004564214528