Estimates for the Green's function of the discrete bilaplacian in dimensions 2 and 3

We prove estimates for the Green's function of the discrete bilaplacian in squares and cubes in two and three dimensions which are optimal except possibly near the corners of the square and the edges and corners of the cube. The main idea is to transfer estimates for the continuous bilaplacian using a new discrete compactness argument and a discrete version of the Cacciopoli (or reverse Poincar\'e) inequality. One application that we have in mind is the study of entropic repulsion for the membrane model from statistical physics.