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 conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems, first-passage percolation models and minimal cuts in the Z^D lattice with random capacities.
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, in an ``independent'' random environment, 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. In particular, we prove, for all values of n, that the surfaces are delocalized in dimensions d=4 and localized in dimensions d=5. Moreover, the surface delocalizes with power-law fluctuations when d=3 and with sub-power-law fluctuations when d=4. Many of our results are new even for d=1 (indeed, even for d=n=1), corresponding to the well-studied case of (non-integrable) first-passage percolation.
No prior knowledge in the topic will be assumed.
Based on joint work with Barbara Dembin, Dor Elboim and Daniel Hadas. Joint work with Michal Bassan and Shoni Gilboa will also be discussed.
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