First order phase transitions in unbounded spin systemsI: Construction of the phase diagram

The phase diagram and the corresponding infinite volume Gibbs states are constructed for a large class of continuous, unbounded spin models. Our construction relies on a partition of unity mapping our system onto an interacting contour system, a generalisation of Zahradnik's approach to Piragov Sinai theory to interacting contour systems, and a suitable mean field expansion around the minimas of the Hamiltonian.