  We provide a sufficient condition for the uniqueness in distribution of Gibbs
point processes with non-negative pairwise interaction, together with
convergent expansions of the log-Laplace functional, factorial moment densities
and factorial cumulant densities (correlation functions and truncated
correlation functions). The criterion is a continuum version of a convergence
condition by Fern{\'a}ndez and Procacci (2007), the proof is based on the
Kirkwood-Salsburg integral equations and is close in spirit to the approach by
Bissacot, Fern{\'a}ndez and Procacci (2010). In addition, we provide formulas
for cumulants of double stochastic integrals with respect to Poisson random
measures (not compensated) in terms of multigraphs and pairs of partitions,
explaining how to go from cluster expansions to some diagrammatic expansions
(Peccati and Taqqu, 2011). We also discuss relations with generating functions
for trees, branching processes, Boolean percolation and the random connection
model. The presentation is self-contained and requires no preliminary knowledge
of cluster expansions.
