Bill Faris: Connected graphs and tree graphs

This talk compares connected graphs to tree graphs. Expectations of various physical quantities are given by expansions indexed by rooted connected graphs. There are a huge number of such graphs, and their structure is complicated. Rooted tree graphs have a recursive structure that makes them comparatively simple. Furthermore, the expansions for tree graphs have nice convergence properties.

 

There is an explicit term-by-term estimate that shows that rooted connected graph contributions (with negative interactions) are dominated by rooted tree graph contributions (with positive interaction). Thus the expansion for rooted connected graphs has a remarkable cancellation that leads to convergence.