Percolation and Magnetization for Generalized Continuous Spin Models

For the Ising model, the spin magnetization transition is equivalent to the percolation transition of Fortuin-Kasteleyn clusters; this result remains valid also for the conventional continuous spin Ising model. The investigation of more general continuous spin models may help to obtain a percolation formulation for the critical behaviour in SU(2) gauge theory. We therefore study a broad class of theories, introducing spin distribution functions, longer range interactions and self-interaction terms. The thermal behaviour of each model turns out to be in the Ising universality class. The corresponding percolation formulations are then obtained by extending the Fortuin-Kasteleyn cluster definition; in several cases they illustrate recent rigorous results.