The membrane model

The membrane or bilaplacian model was first introduced in the physics literature to model random interfaces with constant curvature, and studied mathematically for the first time by Sakagawa and Kurt. It is a centered multivariate Gaussian whose covariance is given by the discrete bilaplacian operator on the lattice. It is tempting to think of it as a kin of the discrete Gaussian free field (GFF), and indeed many results can be deduced with the same methods for both, as for example the fluctuations of the maximum in higher dimensions. Moreover, as the GFF, it also can be seen as a generalised Gaussian variable arising as scaling limit of discrete models, for example in the odometer of the divisible sandpile or in height fluctuations of spanning forests. We will discuss some of the interesting features of the model and some open conjectures.

The talk is an overview of the area and should be accessible to everybody.

Based on joint works with Alberto Chiarini, Rajat Subhra Hazra and Wioletta Ruszel.