Geodesic networks and the disjointness gap in the directed landscape

TL;DR

This paper explores the disjointness gap in the directed landscape, revealing how it encodes exceptional geodesic networks and relates to Busemann functions, with proofs based on coalescent geometry properties.

Contribution

It establishes a novel correspondence between the disjointness gap and geodesic networks in the directed landscape, advancing understanding of its geometric structure.

Findings

Disjointness gap fully encodes exceptional geodesic networks.

Zeroes and local minima of the gap characterize network features.

Results are deterministic given coalescent geometry properties.

Abstract

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. In this paper, we show 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. We give a similar correspondence relating semi-infinite geodesic networks to a Busemann gap function. The proofs are deterministic given a list of soft properties related to the coalescent geometry of the directed landscape.