On the Gibbsian nature of the random field Kac model under block-averaging

We consider the Kac-Ising model in an arbitrary configuration of local magn etic fields eta = (etai)(i is an element ofZ)(J),in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormali zed' infinite volume measures obtained by block averaging any of the Gibbs- measures corresponding to fixed eta, with block-length small enough compare d to the range of the Kac-interaction. We show that these measures are Gibb s measures for the same renormalized interaction potential. This potential depends locally on the field configuration eta and decays exponentially, un iformly in eta, for which we give explicit bounds. The construction of the potential is based on a high temperature-type cluster expansion.