The metric removability of interfaces in the directed landscape

TL;DR

This paper investigates whether the geometry of the directed landscape can be reconstructed from certain curves, showing that interfaces are removable while geodesics are not, with implications for understanding the landscape's structure.

Contribution

It demonstrates that interfaces in the directed landscape are metric removable, whereas geodesics are not, revealing fundamental differences in their geometric roles.

Findings

The set of times where geodesics intersect interfaces has zero dimension.

Interfaces are metric removable in the directed landscape.

Geodesics contain non-trivial information about the landscape.

Abstract

The directed landscape is a prominent model of random geometry which is believed to be the universal scaling limit of all planar random geometries in the Kardar-Parisi-Zhang universality class. It comes equipped with a few different natural simple curves associated to it, such as geodesics and interfaces. Given such a curve, one might wonder whether the geometry off this curve determines the entire landscape, or if in fact, there is non-trivial extra information actually present "on" the curve. In this paper, we show that the former is true for an interface in the directed landscape, while the latter is true for a geodesic instead. Further, as is used in the proof of the first assertion above, we show that the set of times where any geodesic intersects an interface a.s. has dimension zero.