We study scaling limits and corresponding large deviation
principles of random fields perturbed by an attractive force towards the origin and/or by hard-wall (wetting) constraints. In particular, we analyse the critical situation that the rate function admits more than one minimiser leading to a concentration of measure problems.
Our models are in fact interface models with Laplacian interaction, and such linear chain models with Laplacian interaction appear naturally in the physics literature in the context of semi-flexible polymers. We discuss these connections as well as the ones with the related gradient models. These random fields are a class of model systems arising in the studies of random interfaces, critical phenomena, random geometry, field theory, and elasticity theory.