A general coupling for Ising models and beyond

The couplings between the Ising model and its graphical representations, the random-cluster, random current and loop O(1) models, are put on common footing through a generalization of the Swendsen-Wang-Edwards-Sokal coupling. A new special case is that the single random current with parameter 2J on edges of agreement of XOR-Ising spins has the law of the double random current. The coupling also yields a general mechanism for constructing conditional Bernoulli percolation measures via uniform sampling, providing new perspectives on models such as the arboreal gas, random d-regular graphs, and self-avoiding walks. As an application of the coupling for the Loop-Cluster joint model, we prove that the regimes of exponential decay of the q-state Potts model, the random-cluster representation, and its q-flow loop representation coincides on the torus, generalizing the Ising result. By providing a new and equivalent definition of the plaquette random cluster model, we are able to generalize the loop–cluster coupling to lattice gauge theories.