Bernoulli Society: One World Probability Webinar #2

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• Barbara Dembin (Université de Strasbourg),
• Ron Peled (University of Maryland)

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Minimal surfaces in a random environment

Abstract

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 (ground states of the random-bond Ising model).

We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. The basic objects of study are the typical fluctuations of the surface and its energy. We first discuss these in the context of first-passage percolation and the random-bond Ising model where we present recent progress and highlight the many remaining challenges. We then specialize to a recently introduced model, that we term harmonic MSRE, in an “independent" or "Brownian” random environment, whose special properties have allowed to push the mathematical frontiers. Among the new results are precise determination of fluctuation exponents (Brownian case), or new bounds (independent case), as well as scaling relations that tie these together.

Based on joint works with Michal Bassan, Paul Dario, Dor Elboim, Shoni Gilboa and Daniel Hadas.