Estimates for the Green's function of the discrete bilaplacian in dimensions 2 and 3
This provides rigorous mathematical foundations for analyzing the membrane model, a problem in statistical physics, by establishing sharp bounds on the discrete bilaplacian Green's function.
The authors prove optimal estimates for the Green's function of the discrete bilaplacian in 2D and 3D domains, with the main application being the study of entropic repulsion for the membrane model in statistical physics.
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 Poincaré) inequality. One application that we have in mind is the study of entropic repulsion for the membrane model from statistical physics.