  The presence of a random magnetic field in ferromagnetic systems leads, in
the broken phase, to an anomalous $O(\sqrt{1/N})$ convergence of some
thermodynamic quantities to their asymptotic limits. Here we show a general
method, based on the replica trick, to compute analytically the $O(\sqrt{1/N})$
finite size correction to the average free energy. We apply this method to two
mean field Ising models, fully connected and random regular graphs, and compare
the results to exact numerical algorithms. We argue that this behaviour is
present in finite dimensional models as well.
