A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations. Our results agree with predictions from the physics literature.
Joint work with Barbara Dembin, Dor Elboim and Daniel Hadas.