The KPZ (Kardar-Parisi-Zhang) universality class is a family of one-dimensional random growth models and two-dimensional random metric and polymer models that are conjectured to exhibit the same universal scaling behaviour. A central object in the subject is the directed landscape, a random two-dimensional geometry whose optimizing paths, or geodesics, coalesce and form intricate random networks.

In this talk, I will present a new coupling between its fixed-time marginals, known as the Airy sheet, and the Airy line ensemble, the edge-scaling limit of Dyson Brownian motion. I will then show how this coupling can be used to study geodesic coalescence and give a particularly short construction of the Airy sheet.

Finally, I will show how the same ideas extend to discrete polymer models and the KPZ equation.

This is joint work with Duncan Dauvergne.