Two-dimensional Gibbsian point processes with continuous spin-symmetries

We consider two-dimensional marked point processes which are Gibbsian with a two-body-potential of the form U = JV +K, where J and K depend on the positions and V depends on the marks of the two particles considered. V is supposed to have a continuous symmetry. We will generalise the famous Mermin-Wagner-Dobrushin-Shlosman theorem to this setting in order to show that the Gibbsian process is invariant under the given symmetry, when instead of smoothness conditions only continuity conditions are assumed. We will achieve this by using Ruelle’s superstability estimates and percolation arguments.