Stability for a continuous SOS-interface model in a randomly perturbed periodic potential

Abstract

We consider the Gibbs-measures of continuous-valued height configurations on the $d$-dimensional integer lattice in the presence a weakly disordered potential. The potential is composed of Gaussians having random location and random depth; it becomes periodic under shift of the interface perpendicular to the base-plane for zero disorder. We prove that there exist localized interfaces with probability one in dimensions $d\geq 3+1$, in a `low-temperature' regime. The proof extends the method of continuous-to-discrete single- site coarse graining that was previously applied by the author for a double-well potential to the case of a non-compact image space. This allows to utilize parts of the renormalization group analysis developed for the treatment of a contour representation of a related integer-valued SOS-model in [BoK1]. We show that, for a.e. fixed realization of the disorder, the infinite volume Gibbs measures then have a representation as superpositions of massive Gaussian fields with centerings that are distributed according to the infinite volume Gibbs measures of the disordered integer-valued SOS-model with exponentially decaying interactions.