On forest expansions for two-body partition functions on tree-like interaction graphs

We study tree approximations to classical two-body partition functions on sparse and loopy graphs via the Brydges–Kennedy–Abdessalam–Rivasseau forest expansion. We show that for sparse graphs (with large cycles), the partition function above a certain temperature T∗ can be approximated by a graph polynomial expansion over forests of the interaction graph. Within this region, we show that the approximation can be written in terms of a reference tree T on the interaction graph, with corrections due to cycles. From this point of view, this implies that high-temperature models are easy to solve on sparse graphs, as one can evaluate the partition function using belief propagation. We also show that there exist a high- and low-temperature regime, in which T can be obtained via a maximal spanning tree algorithm on a (given) weighted graph. We study the algebra of these corrections and provide first- and second-order approximation to the tree Ansatz, and give explicit examples for the first-order approximation.