Consider the lattice Z^2, and assign length 1 or 2 to every edge by flipping a series of independent fair coins. This gives a random weighted graph, and looking at distances in this graph gives a random planar metric. This model is expected to have a continuum scaling limit as we decrease the spacing between lattice points. Moreover, most natural models of random planar metrics and random interface growth (the so-called `KPZ universality class') are expected to converge to the same limiting geometry. The goal of this talk is to introduce this limit, known as the directed landscape, and describe at least one model where we can actually prove convergence.

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.