Cluster expansions: Necessary and sufficient convergence conditions

TL;DR

This paper establishes a new, comprehensive convergence criterion for activity expansions in equilibrium statistical mechanics, applicable to systems with various pair potentials, and improves existing conditions for certain models.

Contribution

It introduces a novel convergence condition involving a sign-flipped Kirkwood-Salsburg operator, unifying and extending previous criteria, and applies it to hard-core and polymer systems.

Findings

New necessary and sufficient convergence condition for correlation functions.

Recovery of known criteria like Kotecký-Preiss and Fernández-Procacci.

Improved convergence conditions for hard-core and polymer systems.

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