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We compare the different convergence criteria available for clusterexpansions of polymer gases subjected to hard-core exclusions, with emphasis onpolymers defined as finite subsets of a countable set (e.g. contour expansionsand more generally high- and low-temperature expansions). In order ofincreasing strength, these criteria are: (i) Dobrushin criterion, obtained by asimple inductive argument; (ii) Gruber-Kunz criterion obtained through the useof Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us viaa direct combinatorial handling of the terms of the expansion. We show that forsubset polymers our sharper criterion can be proven both by a suitableadaptation of Dobrushin inductive argument and by an alternative --in fact,more elementary-- handling of the Kirkwood-Salzburg equations. In addition weshow that for general abstract polymers this alternative treatment leads to thesame convergence region as the inductive Dobrushin argument and, furthermore,to a systematic way to improve bounds on correlations.

On the convergence of cluster expansions for polymer gases