High points for the membrane model in the critical dimension

In this notice we would like to study the fractal structure of the set of high points for the membrane model in the critical dimension d=4. We are able to compute the Hausdorff dimension of the set of points which are atypically high, and also that of clusters, showing that high points tend not to be evenly spread on the lattice. We will see that these results follow closely those obtained by Olivier Daviaud for the 2-dimensional discrete Gaussian Free Field.