Fock Spaces, Fermi Fields, and Applications

After introducing the symmetric Fock space structure associated with the free field, we define the fermionic field from creation and annihilation operators on the skew symmetric Fock space, and show how it can be used to prove versions of the transfer current theorem for rooted spanning forests and for spanning trees. Applications are given in the case of the complete graph, in particular to prove the existence of a giant subtree and study its renormalization. We also establish the supersymmetry relation which the corresponding bosonic field, which is identified with the free field. We finally give an example of an interaction between trees and loops, which can be represented by a local interaction between the bosonic and fermionic fields.