We study scaling limits and corresponding large deviation principles of integrated random walks perturbed by an attractive force towards the origin. In particular we analyse the critical situation that the rate function admits more than one minimiser leading to concentration of measure problems. The integrated random walk models are in fact interface models with Laplacian interaction, and such linear chain models with Laplacian interaction appear naturally in the physical 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. If time permits we outline basic techniques and give an overview about future research projects in this area.