Cluster Expansions: Necessary and Sufficient Convergence Conditions

Abstract

We prove a new convergence condition for the activity expansion of correlation functions in equilibrium statistical mechanics with possibly negative pair potentials. For non-negative pair potentials, the criterion is an if and only if condition. The condition is formulated with a sign-flipped Kirkwood–Salsburg operator and known conditions such as Kotecký–Preiss and Fernández–Procacci are easily recovered. In addition, we deduce new sufficient convergence conditions for hard-core systems in mathbb {R}^d and mathbb {Z}^d as well as for abstract polymer systems. The latter improves on the Fernández–Procacci criterion.