Toronto Probability

The directed landscape is a random directed metric on the plane that arises as the scaling limit of metric models in the KPZ universality class. For a pair of points p, q, the disjointness gap G(p; q) measures the shortfall when we optimize length over pairs of disjoint paths from p to q versus optimizing over all pairs of paths. Any spatial marginal of G is simply the gap between the top two lines in an Airy line ensemble. It turns out that when the start and end time are fixed, the disjointness gap fully encodes the set of exceptional geodesic networks. The correspondence uses simple features of the disjointness gap, e.g. zeroes, local minima. The proofs are deterministic given a list of soft properties related to the coalescent geometry of the directed landscape. We will look at a similar correspondence relating semi-infinite geodesic networks to a Busemann gap function, time permitting.