Applications of Cluster Expansion

The main topic of this thesis is the cluster expansion technique and its applica- tions to a variety of problems ranging from probability to physics and chemistry. The thesis is divided into a first part of relevant known results from the literature, and a second part where we present our contribution. We start by recalling some central aspects of the cluster expansion, and hence, general cluster expansion theorems in the grand-canonical and canonical ensem- bles and related results. Then, we present a classical problem in probability about computing large and moderate deviations as well as its formulation in statistical mechanics in the canonical/micro-canonical and the canonical/grand-canonical ensembles. We consider both the case of continuous - in R^d - and discrete - in Z^d - systems of interacting particles. In the second part, we present our results. First, we consider a system of classical particles confined in a box Λ ⊂ Rd with zero boundary conditions in- teracting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime, we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates compar- ing with the infinite volume versions of the above are also provided. Second, we show the validity of the cluster expansion in the canonical ensemble for the Ising model. We compare the lower bound of its radius of convergence with the one computed by the virial expansion working in the grand-canonical ensemble. Us- ing the cluster expansion we give direct proofs with quantification of the higher order error terms for the decay of correlations, and also in this case, for central limit theorem and large deviations. In the last part of the thesis, using a strategy given in the literature in the grand-canonical ensemble, we perform the cluster expansion for colloids in the canonical ensemble, considering periodic boundary conditions. The novelty consists in the fact that we establish a hierarchy in the order of integration, which allows to work with the effective system.