High points for the membrane model in the critical dimension

TL;DR

This paper investigates the fractal structure of high points in the critical 4-dimensional membrane model, computing their Hausdorff dimension and revealing their non-uniform distribution, similar to results for the 2D Gaussian Free Field.

Contribution

It provides the first analysis of the fractal geometry of high points in the 4D membrane model, extending techniques from the 2D Gaussian Free Field.

Findings

Hausdorff dimension of high points computed

High points are unevenly distributed on the lattice

Results align with those for the 2D Gaussian Free Field

Abstract

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.