The 27 geodesic networks in the directed landscape

The directed landscape is a random directed metric on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. Typical pairs of points in the directed landscape are connected by a unique geodesic. However, there are exceptional pairs of points connected by more complicated geodesic networks. We show that up to isomorphism there are exactly $27$ geodesic networks that appear in the directed landscape, and find Hausdorff dimensions in a scaling-adapted metric on $\mathbb R^4_\uparrow$ for the sets of endpoints of each of these networks.