One set of results concerns the model of first-passage percolation. In this model one takes the usual Euclidean space and modifies it by giving weights to small regions, making them "heavier" or "lighter" in a random manner, independently between different regions. The goal is to study how this perturbation of the space affects its geometry. For instance, what is the new shortest path between two points, where the length of a path is the sum of the weights of the regions that it passes through? How far is this new shortest path from the straight line path?
It has long been conjectured that the perturbed geometry should contain "highways". By this we mean that the shortest paths between many starting and ending points would tend to share a common section (a "highway") which is a path which is "lighter" than other paths in its environment and thus serves as an attractor. Some of our main results offer mathematical proof that this picture is correct, with accompanying quantitative estimates.
Beyond the study of shortest paths in the perturbed environment, one may also consider minimal surfaces - the surfaces with given boundary which pass through the regions of space with the smallest sum of weights. Another set of results provides quantitative descriptions of the behavior of such minimal surfaces in the random environment, putting several predictions from the physics literature on rigorous mathematical ground.
A third set of results pertains to the quantum mechanical behavior of partices, as captured by the behavior of random band matrices.
Lastly, results are obtained on the high-density arrangement of square-like molecules with centers on the square lattice. These results relate to the field of liquid crystals and provide rigorous justification to the formation of columnar order.
