A Gaussian interface model with Laplacian interactions in the critical dimension

We consider the real-valued Gaussian field on the $d$-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. Such a field can be interpreted as a model for a d-dimensional interface in d+1-dimensional space. For the model we consider, d=4 is critical in the sense that in higher dimensions, the infinite volume Gibbs measure exists, but not in d=4 and below. Understanding the model requires good estimates on the Green's function of a discrete biharmonic boundary value problem. In this talk, I will present the analytical and probabilistic methods we use to address this problem, and hint at how these results are used to investigate the effect of a 'hard wall' on the interface, requiring the field to be positive inside a certain region.