A generalized version of Groeneveld’s convergence criterion for the virial expansion and generating functionals for weighted two-connected graphs is proven. This criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions ρ_s (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood–Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion; however, a Möbius inversion on the lattice of set partitions enters the derivation of the recurrence relations.

中文翻译：

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回顾Groeneveld的病毒式扩张方法

证明了Groeneveld收敛性准则的一般化版本，该准则适用于加权二连通图的病毒扩展和生成函数。此标准适用于非均匀系统和产量界的相关函数的密度扩展ρ_小号（又名分布函数或阶乘矩措施）与成对的交互宏伟规范吉布斯措施。该证明基于与相关函数的Kirkwood-Salsburg积分方程有关的图形权重的递归关系。证明不使用密度-活性展开的倒数；然而，集合分区格上的莫比乌斯反演进入了递归关系的推导。