Phase structure of two-dimensional spin models and percolation

For a class of classical spin models in 2D satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially. The class contains discrete and continuous spin systems with Abelian and non-Abelian symmetry groups. For the discrete models our results imply that they show either long-range order or are in a soft phase characterized by powerlike decay of correlations; for the continuous models only the second possibility exists. The continuous models include a version of the plane rotator [O(2)] model; for this model we rederive, modulo two conjectures, the Fröhlich-Spencer result on the existence of the Kosterlitz-Thouless phase in a very simple way. The proof is based on percolation-theoretic and topological arguments.