Loop interactions and their representations inFock space

It has been observed that on a weighted graph, an extension of Wilson's algorithm provides an independent pair pT , Lq, T being a spanning tree and L a Poissonian loop ensemble. This association can be interpreted in the framework of symmetric and skew symmetric Fock spaces. Given a weighted graph, we show how to define two types of natural interactions which correspond to local interactions between two Fock spaces. The first type of interaction involves loop ensembles and spanning trees. The second type of interaction involves loop holonomies and random connections.