Lower bounds for the width of domain walls in the random-field Ising model

Using the replica trick we construct a "Mermin-Wagner-Hohenberg" style inequality which places a lower bound on the width of domain walls in the random-field Ising model (RFIM). We apply our inequality to two competing interface models of the RFIM, that of Pytte, Imry, and Mukamel and that of Grinstein and Ma which yield different values of $d_c$, the lower critical dimensionality, and the width of domain walls. If we assume replica symmetry, our lower bound is consistent with the work of Pytte et al. but inconsistent with that of Grinstein and Ma. Our result suggests that $d_c$ for the RFIM is 3. However, this result is not conclusive given our assumption of replica symmetry, and the validity of the inequality as the number of replicas tends to zero. Indeed, our result suggests at least indirectly that consideration of these questions may be essential to understanding the conflicting results obtained from the two interface models.