The Cluster Expansion in Statistical Mechanics: Holder inequality

In this review, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by finite range potentials. The Hamiltonian $H_{0}+V$ need be stable in the extended sense that
$H_{0}+4V+KN\geqq{0}$ form some $K$. In this situation, with a mild technical condition on the potentials, the cluster expansions converge and infinite volume limit of the correlation functions existes, at low enough density. These infinite volume correlation functions cluster exponentially. Following the usual literature, we define a class of interacting boson and fermion particle theories with a matter-like potential, $1/r$ suitably truncated at large distance. This system would collapse in the absence of the exclusion principle. The potential is unstable, but the Hamiltonian is stable. This provides an example of a system for which this is method proves existence of the infinite volume limit, that is not covered by the classic work of Ginibre, which requires stable potentials. The main focus of this review is to discuss a key ingredient, a type of Holder inequality for the expectation values of spatially smeared Euclidian densities, a special interpolation theorem. The cluster expansion as developed here is purely a geometric analysis of the paths that realize the traces in path space. The total path space integral is split into subsets in which paths avoid certain regions and must hit other regions.