Revisiting Groeneveld’s approach to the virial expansion

Journal of Mathematical Physics, Volume 62, Issue 2, February 2021. 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 ρs (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.