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We study two free energy approximations (Bethe and plaquette-CVM) for theRandom Field Ising Model in two dimensions. We compare results obtained bythese two methods in single instances of the model on the square grid, showingthe difficulties arising in defining a robust critical line. We also attemptaverage case calculations using a replica-symmetric ansatz, and compare theresults with single instances. Both, Bethe and plaquette-CVM approximationspresent a similar panorama in the phase space, predicting long range order atlow temperatures and fields. We show that plaquette-CVM is more precise, in thesense that predicts a lower critical line (the truth being no line at all).Furthermore, we give some insight on the non-trivial structure of the fixedpoints of different message passing algorithms.

Random Field Ising Model in two dimensions: Bethe approximation, Cluster Variational Method and message passing algorithms