# The maximum of the four-dimensional membrane model

## TL;DR

This paper establishes the convergence in distribution of the maximum of the four-dimensional membrane model on large boxes, using sharp Green's function estimates for the discrete Bilaplacian.

## Contribution

It provides the first sharp estimates for the Green's function of the discrete Bilaplacian in four dimensions, crucial for proving the maximum's convergence.

## Key findings

- Convergence in distribution of the maximum of the 4D membrane model.
- Sharp Green's function estimates for the discrete Bilaplacian.
- Potential applications of these estimates in related models.

## Abstract

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 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.