  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.
