Joint work with E. Bolthausen (University of Zurich) and N. Kurt (TU Berlin).

In this talk we are interested in studying the behavior of a d-dimensional interface around its phase transition. We will specifically discuss the case of a model of "pinning''. This problem arises when one considers an interface rewarded every time it touches the 0-hyperplane. Then there is a competition between attraction to the hyperplane and repulsion due to the decrease of entropy for interfaces pinned at 0. Tuning the strength of the attraction, two behaviors are possible: either energy wins, and the interface stays localized close to 0, or entropy wins, and the interface is repelled away from 0. We will study the effects of pinning for a particular effective interface, the membrane or Bilaplacian model, which is akin to the discrete Gaussian free field. We will draw a parallel between the two models and show how in the membrane case a positive pinning strength localises the field in higher dimensions.