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 Koteck$\acute{y}$-Preiss and Fern$\acute{a}$ndez-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 Fern$\acute{a}$ndez-Procacci criterion.

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Cited by in corpus (1)

• Cluster expansion for a dilute hard sphere gas dynamics