Cluster expansions: A review of the main strategies and their relations.

Abstract:  Some recent improvements on the convergence criteria for cluster expansions has motivated us to review the different strategies to prove convergence.  The different aree based on four different mathematical tools: (1) tree-graph bound, (2) Kirkwood-Salsburg equation, (3) inductive approach, and (4) Penrose identity.  I will first review the notion of cluster expansion and some applications and then present a comparative discussion of the different summability approaches.