Quantum ergodicity ofC * dynamical systems

We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with supperstable interaction.To the memory of Roland DobrushinResearch partially supported by the Isaac Newton Institute Cambridge.Research supported by the Swedish Natural Science Research Council and the Deutsche Forschungsgemeinschaft.