Long range correlations and phase transition in non-equilibrium   diffusive systems
  We obtain explicit expressions for the long range correlations in the ABC model and in diffusive models conditioned to produce an atypical current of particles.In both cases, the two-point correlation functions allow to detect the occurrence of a phase transition as they become singular when the system approaches the transition. 
T. Bodineau, B. Derrida, V. Lecomte, F. van Wijland