Cluster Expansions & Correlation Functions

Cluster expansions were introduced at the dawn of Statistical Mechanics for the study of high temperature gases of interacting particles. They constitute a powerful perturbative method that is suitable for geometrically large systems, such as encountered in Statistical Physics. Numerous articles have contributed to the subject; a list of relevant publications is [GK, Cam, Bry, KP, Pfi, Dob, BZ, Mir] and references therein. Cluster expansions can now be found in standard books, see Chapter 4 of Ruelle [Rue], or Chapter V of Simon [Sim]. They apply to continuous systems such as classical or quantum models of interacting particles, and also to discrete systems such as polymer models, spin models (with discrete or continuous spin spaces), or lattice particle models. The methods for treating these various situations share many similarities, but a somewhat different cluster expansion was so far required in each case. The exposition of the cluster expansion is often intricate, and the paper of Kotecký and Preiss [KP] should be singled out for proposing a clear and concise theorem, that involves a neat criterium for the convergence of the expansion. This theorem applies to discrete systems only, and its rather difficult proof was subsequently simplified in [Dob, BZ, Mir]. Explicit expressions for the contribution of cluster terms can be found in the article of Pfister [Pfi]; this helps clarifying the situation and allows for computations of lowest order terms. The condition for the convergence is the same as in [KP] in the case where polymers are subsets of a lattice. The goal of this paper is to present a general theorem that applies to both continuous and discrete systems. The condition for the convergence is given by an extension of the criterium of Kotecký and Preiss, see Equation (3) below, and the contribution of cluster terms involves explicit expressions. Furthermore, we derive expressions and estimates for correlation functions. The theorems presented here are illustrated in Section 4 in three different situations, namely classical and quantum gases of interacting particles, and lattice polymer models.