Analyticity and mixing properties for Random Cluster Model with q> 0 on Z d Aldo Procacci Departamento de Matem´atica UFMG 30161-970 - Belo Horizonte - MG Brazil Benedetto Scoppola Dipartimento di Matematica Universit´ a “Tor Vergata” di Roma V.le della ricerca scientifica, 00100 Roma, Italy April 4, 2006 Abstract We study the Random Cluster Model on Z d for p near either 0 or 1 and for all q> 0 and we prove by mean of cluster expansion methods the analyticity of the pressure and finite connectivities in both regimes. These results are valid also in the regime q< 1 and they imply that percolation probability is strictly less than 1. 1 Introduction The Random Cluster Model (RCM) is a stochastic process introduced in the early ’70 by Fortuin and Kastelyn [4] which had a very relevant impact in probability and statistical mechanics. The process is initially defined on a finite graph. Each edge of the graph can be open or closed, and the process depends on two parameters, namely p and q, representing respectively the weight of each open edge of the graph and the weight of each connected component of open edges of the graph. The process is then defined on a countably infinite graph, by studying the limit of suitably chosen sequences of finite sub-graphs with suitably chosen boundary conditions. Varying the parameters p and q some of the most popular systems in statistical mechanics (e.g. Ising and Potts model) and probability (e.g. Bernoulli percolation) may be recovered. The RCM has been mainly investigated when the underlying graph is the regular cubic lattice Z d , but during the last decade a growing interest about RCM and related statistical mechanics systems on general graphs has emerged. Few results on RCM can be proved for all the values of the parameters q and p. In particular, the existence of the pressure, its independency on boundary conditions and its differentiability have been proved in [6] for Z d , and for a certain class of general graphs in [10]. This shows that the whole machinery of the statistical mechanics, and its probabilistic counterpart, can be used for all the values of the parameters of the RCM. However the study of the statistical mechanics properties of RCM has been developed so far only in the region q ≥ 1 where the powerful tool given by the so-called FKG inequalities is available. In particular, by comparison inequalities (see [4], [1] and [5]), is possible to prove that, for q ≥ 1, it exists a critical value 1