Real space renormalization group analysis of the random field Ising model

We present an analytic Migdal-Kadanoff renormalization group analysis of the random field Ising model. The renormalization flows to a zero-temperature critical point, from which we calculate independently three critical exponents in arbitrary dimension. In three dimensions the magnetization exponent beta approximate to 0.02, and the Schwartz-Soffer inequality is almost satisfied as an equality. Expanding analytically in epsilon = d - 2 we find that a and the distance from the upper bound of the equality go to zero exponentially with 1/epsilon(2).