Bill Faris: Cluster expansions applied to cluster expansions

Abstract: In many-body physics it is common to encounter series expansions whose terms are indexed by graphs, more precisely, by weighted colored graphs. The vertices of a graph are particles, the colors are the locations of the particles, with each location there is an activity parameter. The interaction weights are associated with the edges.  When the weights are negative there is a remarkable convergence result for the sum over rooted connected graphs. This sum gives the expected number of particles at the root location.

A sum over connected graphs is called a cluster expansion. It turns out that the cluster expansion for particles has an application to a cluster expansion for random fields. A location is now a connected graph associated with the field, and the activity is the contribution of the graph. The talk will treat a particularly simple random field for which the calculations are (almost) transparent.