We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green’s function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. d=4 is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.
"Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension." Ann. Probab. 37 (2) 687 - 725, March 2009. https://doi.org/10.1214/08-AOP417