Abstract: The discrete membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is a very close relative of the discrete Gaussian free field (DGFF), for which instead the most likely interfaces are those preserving the mean height. However working with the two models presents some key differences, in that in the MM the shape is driven by the biharmonic operator, while the DGFF is essentially a Gaussian perturbation of harmonic functions. In particular, a lot of tools (electrical networks, random walk representation of the covariance) are available for the DGFF and lack in the MM. In this talk we will review some basic properties of the MM, and we will investigate a random walk representation for the covariances of the MM and what it can bring forth in terms of scaling limits of its extremes.
This talk is based on joint works, partly ongoing, with Biltu Dan, Rajat Subhra Hazra (ISI Kolkata) and Rounak Ray (TU/e).