Study of the order of the phase transition in pure U(1) gauge theory   with Villain action
  We address the question of the order of the deconfinement phase transition of four dimensional U(1) lattice gauge theory. Simulations of the Z-gauge theory dual to the Villain action on toroidal lattices up to lattice sizes of 28^4 give results consistent with both, a vanishing and a nonvanishing discontinuity in the thermodynamic limit. A decision on the order of the phase transition requires still larger lattice sizes. 
J. Cox, T. Neuhaus and H. Pfeiffer