Counting independent sets with the cluster expansion

I'll present two tools from statistical physics, abstract polymer models and the cluster expansion, and show how they can be used in extremal and enumerative combinatorics to give very good approximations to the number of (weighted) independent sets in certain graphs. In one application we use the cluster expansion in concert with Sapozhenko's container lemma (and Galvin's generalization) to obtain new results on the weighted number of independent sets in the hypercube and their typical structure. Joint work with Matthew Jenssen.