Phase Transitions in Multidimensional Long-Range Random Field Ising Models

We extend a recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in a class of ferromagnetic random field Ising models. Our proof combines a generalization of Fr\"ohlich-Spencer contours to the multidimensional setting, proposed by two of us, with the coarse-graining procedure introduced by Fisher, Fr\"ohlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions $J_{xy}=|x-y|^{- \alpha}$ when $\alpha > d$ in dimension $d\geq 3$ if we have a suitable system of contours, yielding an alternative approach that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen suggested that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions.