Scaling limits of geodesics and upper tail asymptotics in the directed landscape
Two central objects in the Kardar-Parisi-Zhang universality class are the Airy process and the directed landscape. The latter can be thought of as the negative of a random directed metric space (i.e., paths are directed and have weights, which are maximized to give geodesics), and the former as a process recording the weight of the geodesic from a fixed starting point to a varying ending point. A natural question is to understand the behavior of these random objects under the upper tail conditioning: namely, we condition on the weight of geodesics between fixed points to be unusually large, or, equivalently, the value of the Airy process to be large at a given point or points. How do they look and what is the probability of such events? We will discuss a recent geometric and probabilistic technique to address these question for the Airy process, related to the tangent method introduced by Colomo-Sportielli and rigorously implemented by Aggarwal in the context of the six-vertex model. We will also discuss a development of these techniques to establish the scaling limit of the geodesic in the directed landscape under this conditioning, confirming a conjecture of Liu and Wang. This is based on joint works with Shirshendu Ganguly and Lingfu Zhang.