Networks, Ising Model, Flows, and Configurations

We show that edge occupation fields have remarkable distributions for intensity 1 (if we consider oriented edges) and 1/2 (if we consider non-oriented edges). Moreover, after conditioning by the vertex occupation field, the loop ensemble of intensity 1/2 defines a remarkable distribution on flows with integral intensity defined on the graph. We also show that after conditioning by the vertex occupation field, the clusters of the loop ensemble of intensity 1 can be used to construct the F–K Ising model, which provides a coupling between this loop ensemble and the real free field. An interpretation of Kramers–Wannier duality is provided in terms of loops winding numbers. A relation is also established between these loops ensembles and a configuration model.