O(2) symmetry breaking vs. vortex loop percolation
  We study with lattice Monte Carlo simulations the relation of global O(2) symmetry breaking in three dimensions to the properties of a geometrically defined vortex loop network. We find that different definitions of constructing a network lead to different results even in the thermodynamic limit, and that with typical definitions the percolation transition does not coincide with the thermodynamic phase transition. These results show that geometrically defined percolation observables need not display universal properties related to the critical behaviour of the system, and do not in general survive in the field theory limit. 
K. Kajantie, M. Laine, T. Neuhaus, A. Rajantie, K. Rummukainen