Combinatorial Species and Cluster Expansions

This paper will survey recent progress on clarifying the connection between enumerative combinatorics and cluster expansions. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. Cluster expansions, on the other hand, are supposed to give convergent expressions for measures on infinite dimensional spaces, such as those that occur in statistical mechanics. There is a dictionary between these two subjects that sheds light on each of them. In particular, it gives insight into convergence results for cluster expansions, including a well-known result of Roland Dobrushin. Furthermore, the species framework provides a context for recent results of Fernández–Procacci and of the author. 2000 Math. Subj. Class. Primary: 60K35, 82B05, 05A15; Secondary: 82B20, 05C30.