In this talk I want to present the connection between the so-called cluster expansion and relation between species of graphs.
Cluster expansion is a way of dismantling highly complex correlation patterns in probability distributions describing the long term behaviour of high dimensional dynamical systems. The terms of these expansions are labeled by species of graphs, like multi-rooted simple or double connected graphs. Changing the expansion parameter is related to functional relations between the generating functions of different species.
Finally, I will discuss how one can prove convergence of theses expansion, which requires to estimate them by an expansion labeled by trees.