We show that the centred maximum of the four-dimensional membrane model on a box of sidelength $N$ converges in distribution. To do so, we use a criterion of Ding, Roy and Zeitouni (Ann. Probab. 45 (2017) 3886–3928) and prove sharp estimates for the Green’s function of the discrete Bilaplacian. These estimates are the main contribution of this work and might also be of independent interest. To derive them, we use estimates for the approximation quality of finite difference schemes as well as results for the Green’s function of the continuous Bilaplacian.
"The maximum of the four-dimensional membrane model." Ann. Probab. 48 (2) 714 - 741, March 2020. https://doi.org/10.1214/19-AOP1372