Geodesics and Minimal Surfaces in a Random Environment

Endow the edges of the ZD lattice with positive weights, sampled independently from a suitable distribution (e.g., uniformly distributed on [a,b] for some b greater than a greater than 0). We wish to study the geometric properties of the resulting network, focusing on the following challenges:

Geodesics (first-passage percolation): Given two vertices in ZD, consider the path with minimal sum of edge weights that connects them. How close is this path to a straight line?

Minimal surfaces: Consider the cube {−L,...,L}D and the minimal cut separating the "upper" and "lower" halves of the boundary of the cube. How flat is this cut? I will give a gentle introduction to these challenges, with emphasis on the many open problems and some of the recently obtained results.

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